Bleeding Out is a device used, mostly in competitive online games, as a means of forcing players to rely upon one another. Basically, if a character suffers a mortal wound they will begin to slowly lose health, often resulting in blood squirting out from the body or causing them to fall down, leading to an eventual death should they not quickly find assistance.
Two great examples of different types of bleeding out are observed in Gears of War games and the 360/Windows game Shadowrun.
In Gears, certain modes like Execution allow a player to continue living after they have lost all their health. Once "killed", they will drop to the ground and have a limited window of time in which another player can assist them, bringing them back to life. They are completely vulnerable in this state with no means of defense, meaning that if they do not communicate with their team and quickly request back up the enemy will most likely finishing them off and eliminate them from the round.
Shadowrun, on the other hand, uses Bleeding Out in a different capacity. When hit from behind with a katana or your ressurctor has died, you will slow begin to spout blood and as a result begin lossing a steady stream of health. You must seek a healing Tree of Life if you wish to sustain yourself for long, though this is a temporary fix, or you can find another team member to cast the resurct spell on you again to stop the bleeding unless they die as well. The key difference between this example and Gears is that you still have all combat options avaliable to you while you are bleeding out. The trade off here is that while you can seek temporary assistance, once you have been killed there is no permanent cure for stopping the Bleeding Out process.