PvP= player-versus-player (though it should really be named faction-versus-faction, because that's the dynamic WoW uses)
PvE= player versus environment
There are two factions in WoW, the Horde (Orcs, Trolls, Tauren, and Forsaken) and the Alliance (Humans, Dwarves, Gnomes, Night Elves). On the PvP server, you can fight anyone of the opposing faction anytime you meet them. Blizzard is still working on ways to structure this, and a punishments/rewards system is in the works. The first bit of it is a penalty to your honor if you kill someone substantially lower level than you. More on this as it develops.
On the PvE server, on the other hand, you won't be attacked unless you want to be. You can't exactly charge into an opposing faction's town (they WILL attack you for that) but you control whether or not another player can attack you.