Well who knew Web games were developing so fast!!?
One of the best online, browser based, FPS is FallenEmpire: Legions.
The Website is www.instantaction.com. Here You can find a few games, but the best one is FEL.
This game is ...mmm...a type of "Tribes" (a game), or quake 4, but this one is awesome.
After entering the website you can directly play as a guest and find out how the game is, and then u can register.
Well it has to be downloaded once through browser itself, but later on it dosnt need any downloading.
So i recommend u to try out this one and if anyone are members, You can send me a friend request. My username is - SSTMFRk
Here are some pics; and for those who dont like pics, can see the trailer here : http://www.youtube.com/watch?v=eVsemgxoHxA