When you "kill" a player, (and you don't actually kill him, you "defeated" him, which you can see in any number of PAX East videos) he's not actually dead; he is in a state called Incapacitation, in which he cannot move, shoot or otherwise perfom any action (buffing, giving out ammo, etc).
While Incapacitated, he can choose to wait for a Medic (which will give an available Medic on your team a mission to come revive you in his Objective Wheel), or "tap out", which will respawn him at a friendly Command Post according to a looping countdown timer (you could wait anywhere from 20(-ish) seconds to only one, depending on when you are Incapacitated in the countdown's loop). If he is a Medic himself, he can take an Ability to revive himself (he gets no XP for succeeding though).
While you are Incapacitated, the enemy also has a few options too: Any enemy player can perform a melee "coup de grace" finishing move that forces you to respawn in the next wave, and they get a lot of XP for doing so. Enemy Operatives can use the ability call "Comm Hack" to interrogate you into revealing your teammates' locations to the entire enemy team for 10 seconds; they can see this mission on their Objective Wheel the moment you are Incapacitated (according to the Container City walkthrough preview), at which point it will be a race between your Medic and the enemy Operative as to who will reach you first.
Players also have a couple Abilities that work solely for Incapaciation: the first is the Universal Ability called "Downed Fire", which lets you fire your secondary weapon at a sever penalty (to accuracy, I believe); the second is the Operative ability called "Cortex Bomb", which apparently turns your head into an IED and explodes when you are Incapacitated.
So, how do you plan to deal with Incapacitation, whether you are the victor or victim?