I was playing on a Minecraft server and a guy was killing other people and suddenly it said he was killed for PvP logging. What does that mean?
"PvP Logging" describes the practice of logging out during PvP when it looks like you might lose. It isn't unique to Minecraft, but occurs in all competitive games where losing in PvP incurs any kind of penalty, even as slight as simply increasing your "death" count, thus reducing your kill-to-death ratio. In some cases (such as in Minecraft) simply closing your client or quitting the game instantly logs you out, even in combat; for others, it's a bit more complicated, but people found ways regardless.
There are a few plugins for Minecraft servers to combat this behaviour, like CombatLog and PvP Tag for Bukkit. They all have the problem that the Minecraft servers can't discern between someone trying to log out to save their hide and their clients simply crashing. As such, those plugins can lead to unfair situations as well, especially if somebody finds a way to crash someone else's client.