What is PvE?
During the creation of the guides I've made, I have used the word "PvE" often. It means Player vs. Environment. Essentially, you are going against bosses that Blizzard has put into the game to be difficult to defeat. They take a while to take down, because every person has to use strategy and skillful thinking to take down the boss. Unlike PvP, PvE bosses aren't actual people. But, they are more difficult to kill.
Dungeons
Doing dungeons is a form of PvE, but it's not the main thing that PvE is about. You do dungeons to gear up to do the real stuff - raids. Dungeons are made of 5 people - 3 DPS, 1 Healer, and 1 Tank. The goal of the dungeon is to defeat every boss in it. Dungeons are raids that are much smaller and much, much, easier. Some dungeons reward Valor Points ( the points you use to purchase powerful PvE gear) and all dungeons reward Justice Points ( the points you use to purchase less powerful PvE gear. )
Raids
Ahh... the glory of raiding. Raids are the real meat and potatoes of PvE. This is where you use that dungeon gear to use, and try to get even better gear in the raids. Raids are made of 10 or 25 players trying to defeat all the bosses in the raid. You can say that raids are bigger dungeons that are much harder to complete. But, you get very nice gear and rewards from them. You can more Valor Points from raiding.