A short list of Jack-Jack's powers portrayed in the short include but are not limited to: shape shifting, matter-eating, anti-gravitation, floating, hovering, flying, line of sight teleportation, molecular phasing through objects, telekinesis, super strength, spontaneous combustion, laser-projecting eyes, and indestructibility.
In "The Incredibles," Kari says she plays Mozart for Jack Jack because "Mozart makes babies smarter." When Jack Jack bursts into flames, the dramatic music in the soundtrack is the "dies irae" from Mozart's Requiem Mass in D minor (K. 626).
The trivia item below may give away important plot points.
When Kari plays Mozart for Jack-Jack, he makes an unusual face and stops moving. It's probable that listening to Mozart is what activated his powers; as it's rumored that listening to Mozart enhances children's brain activity.