During the events of "Gideon Descends", Gideon gains magic powers while in his demonic state as gifted by Bill. After he's reverted back to normal, some of his powers remain with him, although he doesn't rediscover them until later. It's never stated what exactly his powers are by the creator.
Seeing how Robbie's powers relate back to his GF counterpart, it is safe to assume with Gideon's powers are: telekinesis and telepathy. Given his demonic form's enlarged stature, he might also gain some form of super strength. After Gideon's returned to normal, I'd imagine that he either keeps just one power or toned down versions of all of them. As for when he discovers that he still has magic, I'd like to think that it would happen one of two ways; Gideon's power are activated in the heat of a battle and he has a kick-butt moment, or they just randomly activate while he's just chilling with Mabel, Pacifica, and Dipper and they all scream.
If anyone else has any other ideas concerning Gideon's powers, please do share! :)