A quest is a task given by a non-player character to the player, who will be rewarded for their effort with various rewards such as money or items. They form an important aspect during the leveling stage of TERA: The Exiled Realm of Arborea, as one of the main activities of it is completing quests.
Various forms of PVP in TERA include duels (1 vs. 1) and mass PVP like Battlegrounds, deathmatches between raids, Guild versus Guild (GvG) battles and Civil Unrest. If you're interested in duels visit Arena in Velika Outskirts outside Velika North Gate.
Player versus Environment (PvE), is the part of a game where players fight, defeat or are otherwise pitted against the game mechanics itself. PvE can be used to describe dungeons, quests, achievements and more. The only time something is not PvE is when it is PvP.
BAM is an acronym for big-ass monster in TERA. These monsters are occasionally also called boss monsters or elite monsters. Generally the idea of BAM is formalized meme of MMORPGs, where the bigger the monster is, the more powerful he is.