My classmates and I in middle school played the fuck out of Gun Mayhem. You used guns to know each other off of the stage. Different guns had different fire rates, bullet patterns, reload times, and knockback power. Everyone starts with handguns and you can get better guns as pickups throughout the match.
They also introduced me to OurWorld, a pretty cozy social MMO with some fun mini games. Funny shit happened when a friend taught me how to pick up girls on there. One turned out to be a drama queen and another danced on top of a table in her underwear. '
Way before Bennet Foddy got famous with "Getting Over It", there was a flash site with all his smaller projects called Foddy.net. Get On Top became a favorite in my circle. You have to push your opponents head into the ground to score a point. I was an absolute god at this game!
In college, we would sometimes play this FPS called Shell Shockers when we were supposed to be doing work on the computer. It was simple, but the guns handled very well and was an all around good shooter for a free browser game you didn't even need an account for.
We often played GBA games via Game Oldies, a browser emulation site.
I'll add more games as I recall them, but this is what came to mind.