A mutant has power through an accident of genetics. This "random" power places very few constraints on the mutant; you can have any background and any imaginable power.
Sometimes mutations manifest as Super powers, at other times they manifest as an extreme aptitude for some other type of schtick. Common types include Guns, Pulp, Sorcery, or Super schticks.
Mutants in City of Heroes
Mutants are found in the Outcasts gang and as leaders among Longbow. (Unlike the official Canon, I do not consider Council/5th Column to be mutants).
Mutants are often distinguished by an unusual skin color themed to their powers. Fire mutants are orange, earth mutants brown and so on.
Born with their powers and often unprepared for when they emerge during puberty, mutants have a hard time to cope with life. New powers emerging in this period that is harrowing even to ordinary humans are scary, and often force a break with family and friends. Prejudice and government interference (the Might for Right act mostly targeted mutants) have made them suspicious and wary of others. At the same time, organizations such as GIFT and even the more nefarious Skulls, Outcasts and Carnival of Shadows strive to give mutants a place to call their own and the vital training that can turn their powers from freakshow exhibits into powerful supers.
GIFT
The official government-sanctioned program to help habilitate mutants and make them productive members of society. They operate a number of boarding schools providing training, camaraderie and security to young mutants. Here mutants can discover their powers in safety, with little risk of harm to themselves or others. GIFT graduates can often find good employment as using their special talents; many join Longbow or the Paragon Police Department as special operatives.
Skulls
Not a mutant gang per se, the skulls recruit many misfits; some of these are mutants. If their abilities fit the image of the gang they get to train them and can quickly advance in ranks; if they do not they either flee or die.
Outcasts
Not really one gang but many. Before the war, many mutants found living in a "normal" society that did not really want them was just too hard and formed gangs. During the war, these gangs often fought skirmishes with Rikti and other forces, and thus felt that they had "liberated" part of the city. Now they are unwilling to return this small part of the world that is theirs to the powers that made them outcasts.
Carnival of Shadows
The history of the Carnival of Shadows involves some magic, but most of the rank and file are mutants. Many mutants end up in freak shows, and most of these eventually come into the care of the Carnival of Shadows. The carnival leaders train them and give them a sense of community.