Arena = the game released in 2012, formerly known as simply "Mage Wars." All the expansions released so far require an Arena (or simply "Mage Wars") core set.
Academy = 2-zone streamlined game that plays quicker, is more portable, etc. Academy cards are legal in Arena, but not the other way around (probably because many keywords and rules from Arena don't apply in Academy). It's being released in 2-Mage packs that represent a stand-alone game, rather than an expansion to Arena, making it an accessible jumping-in point for new players.
Battlegrounds = special expansions that include new ways to play Arena. Follows Arena rules, is completely compatible with arena base set and expansions. That is, Battlegrounds cards are legal in regular Arena matches and vice versa (as I understand it).
Battlegrounds - Domination = the first of these, coming soon (Origins), featuring modular puzzle-like board, objectives, alternate win conditions, various board setups for different scenarios, environmental effects, etc. See here for more info:
http://www.arcanewonders.com/products/battlegrounds-domination