You have the projectile attack spell, which is always available and costs 80 energy. It's the little fireballs and blasts of lightning and such you see flying around everywhere.

At level 3 and above, units can use the ring spell and the strike spell. The ring spell costs 90 energy, to hurt all nearby people and knock them all away.
The strike spell calls down a pillar of fire/lightning bolt to hurt the enemy and lower his stats, also at the cost of 90 energy.
For both of these spells, the caster is at the center of a pillar of light while casting.

As for level powers... every time a unit gets a level it gets a glowing aura thing. As it progresses these get larger and deeper in color, which is why powerful rads glow black and powerful faws have a faint golden glow. Once a unit is level 3 or higher, it begins to create explosions whenever it lands melee attacks. These explosions are merely graphical. When a unit gains level 4 and any level past level 4, it instantly casts a ring spell for free (to keep high level units in heavy melees from being killed instantly). Theoretically there is no limit to the level a character can get, but there are practical limits. There aren't any special powers past level 4 or so that I remember.