Okay, so a few of these have already been said, but take it to mean that they're really worth checking out.
Tengen Toppa Gurren Lagann - Now, yes, it's a hero-mecha anime and it threw me off at first, but I liked the characters so I stuck with it for a few episodes... and then it transformed (on a galactic scale!). This show is the only mech anime I would recommend watching even if you hate the genre because it doesn't really focus on the mecha as much as it does on the frailty of people, without going all emo (bar that one point, but we can forgive). That and it's ending... It's very worth watching.
Great Teacher Onizuka - It's a bit dated, art-wise, but the title character just draws you in and leaves you blinking at the end, wanting more. It's a great little series (I finished it in one very long, coffee-fueled night) and again, it's not what you expect of its genre (which is... (semi-) modern, slice-of-life, school life). It's not dark, it's not edgy, but it's definitely worth picking up during those times you feel like something a little different (and let's all admit that sometimes you just want something to cleanse the palate.)
Another - A short yet dark and twisted school-life/horror anime that will have you guessing. I didn't find it all that scary, myself, but then I live and breathe horror movies, so... That said, the art style is quite nice, the characters pull you in and there's more than enough mystery and suspense to go around.
Accel World - People have already recommended its predecessor, Sword Art Online, but I think Accel World is a better constructed anime and story. The characters are interesting, the story-line intriguing and the world design dark. Though the ending was a little... odd, I still recommend it. The art style is at times cute and beautiful, at others dark and edgy. Well worth checking out.
D.Gray Man - I have only one major issue with this series - it's incomplete. Sad, but true. The story as it stands is pretty decent but what really makes this anime shine is the characters and designs. At times goofy, at other times rather creepy, D.Gray Man is an interesting anime. The battle scenes are done well and some of them are quite shocking, especially when plot twists are thrown in. It's worth a look at if you have the time and nothing better to watch (I hate investing in something to not complete it, especially in the case of a promising something. :< )
Deadman Wonderland - I've only read the manga to this anime but I assure you that even having only seen that much, it's well worth the watch. (If you care to know, it was a reasonably recent read so I'm waiting for memories to fade a little more before watching. Same with some other anime that I might point out.) That said, if you want a pretty dark and macabre plot-line, look no farther. The art, if it follows the manga style, is quite nice and the characters are both interesting and (with some people) repulsively attractive.
Bubblegum Crisis Tokyo 2040 - I'll admit, this was the first 'dark' anime I ever watched and it got me into the whole adult anime thing. It's dark, it's edgy and it's not bad for being a bit old (mid-90s is when I first watched it). There's a bit of fanservice, if you like breasts, butts and thighs, but there's also battles galore and a futuristic Tokyo (as the title suggests). Also some pretty kick-ass music and battles. Definitely worth a check out.
Wolf's Rain - Beautiful art and fanciful future collide in a story about wolves searching for 'heaven'. There's mystery and mayhem and while there's some fluffy bits, it maintains a serious tone most of the way through the movie. The music is also really good (I have a few tracks - Stray, Heaven's not Enough - on my iPod) and the story and characters pull you along. The one issue with it is that it tries to cover a lot of bases with symbolism all over, but if you can ignore that - and it's easy enough to - it's a really good story. I guarantee at least a few tears along the way, too. (Then again, I'm a bit of a cry-baby, so...)
Ghost Hound - This is an interesting little anime about three boys who follow ghost stories and, well, it's odd but there's enough there to pull you along for the whole series - which is 22 episodes long. There's some interesting art in the series, too, especially at certain parts. I recommend it if you've got an empty slot in your anime watching calendar. I mean, it's decent, but there are a lot better ones out there. Still, worth checking out at least once.
Gosick - This is an interesting recommendation because it's rather split - some people love it, others really hate it. I enjoyed it enough to recommend, though. It has an interesting main plot and while I'm not a big fan of the main characters, they do the job of telling that story well enough that I ended up liking the show. It's basically about a Japanese teen going to school in a foreign country and meeting a mysterious girl at the library, but there's a lot more to it than that. The art is pretty decent, as is the sound direction and character designs.
Gunslinger Girl - This might seem to be a light-hearted story about the main characters and how they interact, and on the outside of the can it looks like a slice of life, but this is one of those anime where appearances are very deceiving. The first deception is that the girls seem pretty, cute, normal girls. That's not the case. At all. There's a lot of psychological and dark themes touched on in the series and it's well worth watching just because of that, but the art design, action scenes and level of acting is of pretty good quality too. The story, such as it is, is interesting and it's really worth watching. Like, really.
Jormungand - The story of a child soldier who joins a band of weapon dealers, led by the enigmatic Koko - this gem of an anime I found by clicking a random name in a list and watching just for the hell of it. I don't think many people actually know of it's existence but it's worth checking out. The characters and story are decent, there's enough interplay and joking to balance the darker reality of violence in the world of large-scale weapon dealing and war. The art is also pretty good.
Mahou Shoujo Madoka Magica - When I mentioned anime with deceiving appearances, this was the first thought in my mind (followed by another recommendation to come). It looks like it should be a light, fluffy anime about cute magic girls in dresses and sweet romantic shenanigans... but it's not. It lies. And that's perfectly okay by me because, god damn, did we get a neat freakin' little bundle of lies right here! Just... watch it. Just do. I don't want to spoil anything about it so I'll just say... watch.
Monster - Another case of only having read the manga so far due to waiting for memories to fade a bit more. Never-the-less, the story and characters are, in a word, amazing. Not surprising as they come from the great mind of Urasawa Naoki - the mangaka of Pluto, 20th Century Boys and 21st Century Boys, all great tales about characters. As a character-driven story, the main cast are what drive the story forward and they do so very well. I would recommend reading the manga instead of watching the anime, myself, as I'm not sure what changes were made in the production (and Pluto, which I also recommend, is only available in manga) of the anime. It's a story with very dark themes that deals with war, brainwashing, psychosis and many more mature aspects of life. Set in Germany and the surrounding areas, it's a very nice glimpse into the post-war years there. The art is very different to typical anime art, opting for a more... realistic portrayal of people, so be warned that it might take a little while to get used to it. That said, it's well worth checking out.
Shigofumi: Letters from the Departed - A very short series that follows the story of Fumika, a seemingly young girl who delivers letters from the dead. It's an interesting little story with a couple of points where you just tear up and the ending story is pretty interesting. The art is also very nice. I wouldn't call it dark and edgy but it does touch on some... touchy themes at times - definitely not a kids anime.
Witch-Hunter Robin - An anime with very nice art and an interesting story, too. The characters are enjoyable and there's enough light parts to at least beat back the darker ones a bit here and there. It focuses on Robin and the team of Witch Hunters that she's been assigned to as they look into increased witch activity. As a whole it's a rather artistic show and well worth checking out. (Yes, I know I say that a lot.)
Detective Loki - If you like a bit of mystery with you Norse gods, then check out this little anime. I'm sad that there's not more of the series as it was very interesting. With a few rather creepy and dark pieces, it's a very interesting take on the mythological and detective genres. The main annoyance is the actual secondary main character - an overly-bubbly, squeaky-voiced, pink-haired ditz. I did grow to like her, but the English dub voices made her... rather hard to listen to for long periods. Although I've not seen it, I recommend the sub (though the voices for the rest of the cast were decent enough.) Art is done in a cutesy style, but hey, Loki is the main character, so...
Princess Mononoke - Alright, so, this is one of the staple anime movies in existence. It's up there with Akira, Perfect Blue and the other Ghibli films as movies an anime buff must see at least once. All the Ghibli films are worth checking out, even the ones that are light-hearted romps, but this one is the one you should see if you had to choose between them (imo - everyone has their own opinion when it comes to Ghibli, though most will say at least watch all of them once ^.^). The art shines in a lot of areas, especially the plot-pivotal and action scenes. The characters are lovable and easy to identify with, while the story itself is great and well-paced. As is to be expected of Studio Ghibli, the character designs are noticeably different from each other and the blend of creepy and heart-warming is very well-balanced. I highly recommend it, even if it doesn't quite fit in the genre you're looking for. Maybe watch it as another palette cleanser (of which I will recommend a few more before the end of this rather long list.)
Princess Tutu - Another of the 'not what it seems' club, this anime pretends to be about a magical girl ballerina when in actuality it's about so much more. It's a little slow to start but once it gets into the good stuff it doesn't stop. There's chuckles along the way to keep your spirits from getting too destroyed but the main story focuses on the characters of a fairy tale who can see outside the 'box', as well as the great darkness that threatens all their existence. It's a little longer than most of the anime I've recommended - two seasons worth - but it only gets better as it goes along. The action scenes are very well done art-wise and interesting to watch when they get more involved. The characters are lovable and you'll definitely cheer on some that you didn't think you would. It's worth watching (I'm also saying that a lot, aren't I?) if you want something a little different. It's... very different, but really neat.
Red Garden - This is an odd one that I've added to the list. When I watched it it seemed a lot shorter than it actually is - running a full season - but apart from that, the art style takes a little getting used to. Another 'not what it seems', set in a very recognisable modern world, it's quite difficult to say whether I actually enjoyed the anime or not. The story is... strange, following the story of four women who awaken some powers that allow them to fight monsters. I think the thing that made me unsure about it was the characters themselves - they're very unsure of themselves and are extremely reluctant to use their powers, even in life and death situations. It makes for an odd feeling anime, but it covers both the dark and edgy categories and it is an interesting anime, all told.
Perfect Blue - More a horror story than anything else, it's one of those 'watch at least once' anime movies. Following the story of a super popular pop-idol who decides to turn actress, it's both dark and creepy, keeping you wondering who is the culprit. The art style is decent. The story itself is strong and the scenes will be remembered even years after watching them. Especially the big resolution scene. Well worth the hour or so to watch.
Right, time for a few suggestions that don't quite fit your requested description, but are worth checking out as 'palette cleansers' or at least for a break from the psychological twists and turns you're probably going to be watching for the next long while. Besides, someone else might find something to watch, too. ^.^
Scrapped Princess - I wasn't sure whether to put this one in the list or not but it's actually a lot better story-wise than it seems. It's a little too... fantasy-esque to fit your required descriptions, but it has a really nice story with characters that you learn to love (though there were a few times I wanted to smack a certain someone >.<; ). That said, there are darker parts of the anime, and, hell, the premise itself is a bit dark if you think about it. It's basically about the princess who was thrown away because of a prophecy saying she'd destroy the world - thus, everyone and their mothers are trying to kill her. That's the basic plot but it keeps up a nice flow and progresses at a good pace. The art is a little dated, but still looks decent. Check it out if you have the time and inclination.
Skip Beat! - Yeah, this is like the total opposite of what you requested but it's a very strong character-driven anime that, while not complete (only one season so far), is well worth checking out if you're in the mood to laugh a little/lot. The story follows Kyoko, a woman scorned by the love of her life, and her desire to get revenge by becoming a bigger and better star than he is. Problem is, she doesn't know the first thing about anything to do with show-biz and, well, telling you too much would be spoiling. I know, it sounds very fluffy and 'ugh' but I guarantee that for a laugh, it's well worth checking out. It helps a lot that Kyoko is not at all a fluffly, star-eyed beauty - she's a real normal girl with only revenge on her mind, to her detriment. Also, the facial expressions and situations... well, just... give it a shot. It's well worth a watch. The art is pretty good, too. ^.^
Tears to Tiara - This one looks like a typical fantasy anime, and on many levels, it is - being based on a game of the same name - but there's something about the story and characters that makes it interesting. The story doesn't go the way you'd expect and there's a lot of interesting in-world lore that makes it worth watching for the ideas presented. It's also very pretty to look at. The only issue I have with it is the pacing which is a bit too fast for my liking. Granted, it was based off a game, and there wasn't enough material for two seasons worth of anime, so they did what they could to squeeze as much as they could into 26 episodes. It's decent and definitely worth checking out if you don't mind a touch of heretical ideology and the like. Sadly, most of the cast is a little forgettable, but it's still a good watch.
Oban Star-Racers - An interesting sci-fi racing genre anime. Usually I prefer horror and psychological or fantasy/adventure but this one caught my eye and I soon found myself watching the whole thing in two sittings. The art style is very fresh and the action scenes, very well done. The story is well-paced and does a good job setting up the plot. The characters - especially the main character, Eva - are easy to love and relate to and you find yourself cheering them on through both their personal battles and the races they take part in. It's cute, but with some decent twists and while it never really strays into dark territory, it does hint at parts sometimes. There's also some good laughs to be had.
Avatar: The Last Airbender - Okay, yes, it's popular. Yes, it's western animation mimicking Japanese animation, but the series is just... ugh, so great! It's a longer series, but well worth watching. The characters are lovable and their story unfolds at a decent pace. There's tear-jerking moments and a lot of laughs, but most of all, there's the feeling of having accomplished something after watching it that just sticks with you. It's one of the best western series out there and well worth the love it receives from those who have watched it. Definitely check it out because you're worth it.
Utawarerumono (The One Being Sung) - One of my favourite anime, it's based loosely on the Water Margin (the same book Suikoden and a lot of other games are based on) and follows the story of an amnesic man who is taken in by a friendly village and eventually ends up rebelling against an empire. There's a harem, giant mecha, beast people, heart-warming relationships and funny moments galore. It's cute and cuddly during character interaction, and yet brutal and bloody during the battle scenes. The animation is very well done, especially the battle segments, and the characters are mostly lovable. You never have to worry about who to cheer for - the anime makes it clear who to love and who to hate. In that aspect it's a bit black and white, though there's some grey through-out when dealing with some enemies. That said, the story is a little rushed in places. I chalk this up to it being based on a game, much like the above Tears into Tiara. Also, the music score is pretty good, with the OP and Closing credits being especially recognisable and catchy.
Fruits Basket - What to say about this anime? It's a light, fluffy anime full of laughs, with darker undertones that only become apparent at the end. It follows Tohru Honda, a young teen who recently lost her mother and is living on her own in a tent. When walking to school one day she meets one of her classmates from the secretive Sohma family and from then on her life changes forever. Well, I make it sound a bit more dramatic than it is, but with an anime brimming with humour and good-intentions (seriously!), a bit of drama fits. Especially in this case. The characters are lovable, fun and funny; the art is cute, well-drawn and bright; music fits very well and the story itself, while mainly episodic, does lead to a few surprises. The main character can be a bit annoying with her naive personality, but she's ultimately lovable and said quirk sets up a lot of great jokes. The characters are the main stars of this series and it's no wonder the show is well-loved - each person is different from each other and funny in their own quirky ways. It's really worth a watch, especially on those days when the world has you down. Guaranteed to bring a smile.
One Piece - Yes, super popular. I know you didn't want to hear about the super popular anime but I have to plead the case of this one. One Piece is a story about adventure. It follows a pirate crew from the start of their career to, I guess (considering it's still being created) the finding of the mythical treasure. It's a long slog. A very, very long slog. And yes, the anime has a lot of trouble with pacing. I'm not going to recommend the anime, however, because of this fact - though it's nice to see manga panels animated. I'm recommending you read the manga. Not only is the pacing set as fast as you can read, but there are certain things that don't make it into the anime (though on the flip side, there are certain anime-only scenes that aren't in the manga). That said, the characters are lovable - you're sure to find at least one of the crew to latch on to. The story presents itself as the typical search for the macguffin, but what really makes One Piece different is the details - and boy, howdy! are there details~ It's one manga that doesn't lose steam, that always has a surprise just waiting around the corner. It makes you laugh (a lot), cry (enough) and care about not just the characters, but the world, story and journey as well. I could talk for hours on the subject but I'll just say this - the pros far outweigh the cons to picking up this series. It's no Bleach or Naruto, it only gets better.
Ouran High School Host Club - The last on my list is by no means the least. Indeed, it's one of my favourite anime and deserves a watch. Ouran is a hilarious comedy about Fujioka Haruhi, the veritable fish-out-of-water poor-man in a rich-mans world. Winning a scholarship to the exclusive Ouran High School, Haruhi ends up meeting the boys of the host club... and can't quite escape their clutches. A lesson in how to make a funny anime touch on serious content without losing the humour, the anime is a bit bizarre in certain ways - jumping from scene to scene between episodes - but over-all a cohesive piece of work that just works. The characters are lovable, the art is pretty and each episode brimming with humour (even during the more serious and dark ones). It's not everyone's cup of tea, but it is worth checking out and seeing if you enjoy it. It's very light-hearted and heart warming. Definitely on my highly recommended list.
Well... that went a bit longer than I thought it would but there you go. My top recommendations. Take them as you will! Hopefully you'll find a few worth your while, though. ^.^
Enjoy~
