For the MCU:
In Marvel: Agents of SHIELD They refer to people as gifted (born), enhanced (created), and for people who have undergone a change via Terrigenesis are referred to as Inhumans. People from other planets who have powers are simply referred to as aliens.
In Avengers: Age of Ultron Captain America refers to Quicksilver and the Scarlet Witch as enhanced, without knowing the origin of their powers.
In Jessica Jones those with powers are referred to both as gifted and as freaks (by the haters in New York).
These doesn't yet appear to be a consistent term for these types of people.
For the Comics:
You have Mutants, who are carriers of the X-gene (Homo Sapien Superior). You also have Eternals, Deviants, Inhumans, Subterraneans, and Homo Mermanus (mer-people). These are the terms for people who are born with powers or the potential to have powers.
Additionally there are people who have been artificially enhanced. These are called Super-Soliders and include Captain Amerca, Luke Cage, Frank Simpson, Peter Parker, and Bruce Banner. That page lists a subcategory for how a Super-Solider was created (Gamma Rays, Weapon X, Oz Formula, Extremis, etc).