Stuff like "Actor 1 used Ice Volley!" or "Actor 3 is suffering from poison." I think you can change it in the engine's code, but I'm not 100% sure. So if you don't mind using plugins, Yanfly's Battle Core let's you change without doing all that digging.
That is the battle log, and you would need to do a script mod to change it.
I've moved this thread to JS/Plugin Support. Please be sure to post your threads in the correct forum next time. Thank you.
Your profile says you primarily use VX Ace, but you posted this question in the MV forum. Which one are you using? The answer will be very different. If you ARE using Ace, I'll need to move the thread again.
It does sometimes get a little embarrassing that I often come here to ask for help with something and seldom have anything to actually show for it. If there is one thing though I can commend myself for, it's for sticking with an idea I've been tinkering with for this long now and gradually making headways. Even though I've been reserved about my progress, I can say I'm farther now than where I initially started.