A killing blow refers to when a player causes damage to a target that results in the target's health dropping to zero.
Certain abilities and effects are only activated if a player gets the killing blow on a monster, regardless of how much damage the player actually caused.
Some monsters and players have abilities they can use that also damage themselves. If the monster or player kills itself with this ability, they would have in effect scored the killing blow on themselves. Thus it is possible for a player to not get the killing blow on a monster or player, even if he or she was the only one attacking them.
Most abilities that require a player to land the killing blow to activate also require that the thing in question to be killed would also give the player experience or honor. This does not mean that the ability does not proc off of things such as PvP guards in enemy towns (which yield neither experience nor honor), it simply means that the level of the person you are attacking must not show up as gray in the standard WoW interface - which usually means the enemy in question can be no lower than 8 levels under that of the player.
In a Battleground, the player list is sorted according to the number of killing blows by default. Although it should be noted, they don't count killing blows on totems, pets, or NPCs.