In the Marvel Universe, mutations most often come in the form of physiological modifications to allow for some capability not normally posessed by Homo sapiens, the ability to affect and control some form of matter or energy, or increased/altered mental abilities such as telekenesis, precognitive abilities, telepathy, teleportation, or the modification of others' mental processes. For example, the Angel has wings which allow him to fly, Pyro can control fire, and Jean Grey is telekinetic and a telepath.
Mutants play a tremendous role in the Marvel Universe. For one thing, some of the most well known superhero teams like the X-Men and X Factor are comprised of mutants and fight to protect mutants and humans. For another, some of the greatest villains like the Sentinels came about in order to eradicate mutants. Furthermore, some of the story arcs occurred because of mutants, like The Morlock Massacre and Days of Future Past. Even still, most mutants' powers do not manifest until they hit puberty, and some mutants with latent powers do not immediately have control over them when they first manifest themselves. Sometimes, especially if the mutation is especially powerful or dangerous, out-of-control incidents occur and further humans' fear of mutants.
Even non-mutant superheroes are affected by mutant persecution as they have a chance of giving birth to mutant children. Mister Fantastic and the Invisible Woman of the Fantastic Four have a son, Franklin Richards, who is a mutant. Franklin's mutation may have come about due to his parents' powers. It is known that many of the genes which cause the particular types of mutation in the Marvel Universe is passed through the parents' genes.
Mutants play a smaller, but still substantial role, in the DC comics universe, where they are known as metahumans. Mutants also are a frquent topic in other comic books, and in many science fiction stories.