Shroud of the Avatar will feature nine schools of magic, eight of which are considered elemental in nature. The elemental schools of magic are1:
There is also a ninth school of magic1:
Each school of magic appears to have a different focus, or purpose — Fire magic being for damaging foes, Water magic being for healing allies, and so on. Note, as well, that the symbol for Moon magic depicts the shattered moon of New Britannia. Details on the distinctions between each school are sparse, at present; see below for summaries of known facts.
A magic sigil depicts the interrelationship of each school of magic (and will presumably be instrumental in accessing Shroud of the Avatar’s magic system). As you can see, the Chaos school seems to serve as the intermediary between the other schools: