Games that had a deep impact on me? Hmmm... Okay, in no particular order (mostly because it would be difficult to put them in order of playing them).
Halo: Combat Evolved
Shooters can have a story? A GOOD story? Okay, I kind of knew this already, but it was still a VERY RARE sight to behold. More than that, the story was so interesting that I'd replayed it 200 or more times with various friends on various difficulties with various self-imposed challenges... Just to try to find all the dialogue. Just to memorize it all. The game itself was just a masterpiece of entertainment as well. I haven't played a shooter campaign so much fun since. Especially one that allowed for split-screen coop. Seriously, that was just awesome at the time.
Perfect Dark 64
Another shooter, yes I know. First shooter I ever played that had a storyline. I liked the storyline (though found it difficult to follow when I first played the game). However, the gameplay mechanics were really fun, and what kept me playing was the highly addictive multiplayer modes. In a time when all of multiplayer was just tacked on "Deathmatch" gameplay where you moved 9000000 miles an hour, it had the gonads to do different things with its gameplay. Change the weapons on the map? Check. Add bots? Check. Teams? Check. Different varied game modes? Check. A challenge mode? Check. Personal stats, awards, and an agent ranking? Check! Different difficulty of bots? Check! Add in cheats or specific game modes? Check! Change win conditions of a match? Check! Pick character models? Check! Alternate fire modes on guns? Check! The possibilities were literally endless and I played multiplayer for 9 DAYS of match time. It was the game that taught me how to fire the Falcon 2 as fast as I could. The Falcon 2 was a pistol that fired as fast as you could pull the trigger (mash the button). Held 8 shots to the clip. Very accurate, almost no bullet spread. I could fire those 8 shots off in roughly 2.5 seconds. Sometimes, much faster if I were properly "in the zone". It's a skill that's served me well in a lot of modern games, including games like Mario Party.
Final Fantasy 6
This is the game that got me into playing RPGs in the first place. The story was engrossing, the characters memorable, and the Magic System a lot of fun. The music was simply amazing as well. If you ever ask me why I got into making RPGs or playing them. This is the game right here.
Chrono Trigger
Not much for me to say here. For me, this is the greatest RPG of all time. Even more, it's the greatest game of all time for me. It has its problems and issues, yes, but it remains one of my favorites to this day. I've done everything there is to be done in the game and it was always worth it. It showed me how much fun combat in a game could be (most RPGs have some pretty dull combat). For that, it had an impact on me. It got me to hate playing games that are "mash attack to win" and "spam the best spell".
Secret of Evermore
I didn't know games could be funny until I played this game. I also didn't know they could be fun to explore and interact with.
Secret of Mana
A game I got into purely for the music and the multiplayer. The intro for this game still holds a soft spot in my heart and I will endlessly hum along with that opening theme every single time I hear it. Why aren't more RPGs multiplayer? Especially couch co-op multiplayer? I miss that!
Final Fantasy X
I know this game gets a lot of hate, but I don't really care. I loved Tidus, I thought he was well-written for being what he was (audience stand-in as well as spoiled rich boy). By God, he even had a very character arc in the game if you were paying attention to him and were invested in the story. Anyway, this game taught me that... So long as you've got fun and interesting characters, you've got yourself a game I can get lost in. The story was interesting, despite being so annoying vague... But, I liked it. I also loved the Sphere Grid. I know, I'm weird. But, I liked that you didn't "level up" so much as you earned the right to move around and unlock stuff on the grid that directly made you much more effective in combat. The crafting system though? That freakin' sucked.
Okay, that's my short list. There's a lot of games that have had an impact on my life, but those are the big main ones. If I listed them all, I'd be here all day long.