X-Men: Destiny lands on consoles (and DS) next week, and in preparation for that, developer Silicon Knights released a series of trailers delving deeper into the story and gameplay.
Players choose to start the game as three premade mutants, as you're no doubt aware by now, but each character's powers can be supplemented by X-genes, allowing players to use the powers of notable X-Men characters. The X-genes you equip will also affect the course of the story, so it isn't a gameplay exclusive feature.
While you can't actually make your own custom mutant, there is a fair share of customization in Destiny, and you'll ultimately choose which side of the war you'd prefer to fight in: with the X-Men or the Brotherhood of Mutants?
X-Men launch vignette
Brotherhood launch vignette