PVE: Wtf!?
Yes, I know you're probably wondering what PVE is! Which is convenient because I am prepared to tell you :)
PVE is a mode of play in which the character’s main goal is to defeat a foe in the environment as opposed to another player (that is called PVP; player versus player <link>). This foe can vary from a boar to a demon to even a dragon and so on. PVE Is the way a person goes about leveling their character; although pre-expansion you could level by turning in battleground tokens for a small amount of experience, this has since been changed. This way of play is also the main focus of end-game content which is called raiding. Essentially PVE is a gathering of friends online, and some rivals, to defeat a challenge set forth by the developers (typically described as an 'encounter', 'event, or 'boss').
There is also PVE servers, which means there is no world PVP without enabling it on your toon so you may experience PVE in peace if you so desire! But keep in mind: if you are on a PVE realm you cannot transfer to a PVP one and if you transfer from a PVP one to a PVE one you will not be able to go back to a PVP server.