Betraying is when a player kills a teammate in multiplayer. This is also known as a Team Killing and Player Killing.
In Halo
Betraying may give you a vehicle or a weapon that the target is operating, or it may cause them to drop an objective, such as an oddball, so you can improve you ranking for that game. However, repeated betraying will give your victim(s) the option of booting you from the game. In most games you also get the penalty of a longer respawn time with each betrayal.