Quake, released by id Software in 1996, was the first shooter rendered in true 3D: real polygonal levels you could look up and down in, with real-time lighting and a soundtrack by Trent Reznor. It was a technical leap so large that the engine powered games for a decade afterwards, and it invented online deathmatch as a mass activity.
The game itself is a grim march through gothic and Lovecraftian dungeons with a shotgun, a nailgun and a grenade launcher. It's faster and darker than Doom, and it still plays superbly.
It was the first shooter with genuine 3D polygonal levels and free look, rather than Doom's 2.5D. It also popularised online deathmatch, mod-making and the game engine as a licensed product.
This page runs a browser port of the shareware/demo content. The full game is sold on Steam and elsewhere.
Yes — it's a 1996 game and runs easily in a browser with nothing to install.