Chaos Mutant
A mutant is a creature with a genetic mutation that makes it differ noticeably from the majority of the species.
When Earth-creatures reproduce, their cells reproduce and have to copy their DNA. Reproduction is imperfect, so there are many error-detecting and error-correcting mechanisms to increase reliability. Of course, those mechanisms are themselves imperfect, so in any given copy operation, something like one in every hundred-million base-pairs is copied incorrectly; additionally, environmental conditions, like exposure to radiation, certain chemicals (called mutagens), or certain viruses, can introduce errors into a cell's chromosome. These changes are mutations.
In the real world, mutations are usually of limited effect. At best, a single mutation might improve the efficiency of some metabolic pathway; most will have no or minimal effect; and at worst, a mutation can cause cancer or other genetic diseases. Over many generations, a species can change quite dramatically, depending on what selective pressures they face in their environment, and what effect mutations have on their bearers' ability to reproduce; this is the process of evolution. That said, the effects are usually not perceptible on the human time-scale.
In fiction, mutations have much more extreme effects, like granting superpowers. This is especially the case in the world of comics and cartoons, as exemplified by groups such as the X-Men: an unknown-to-modern-science "X-gene" (or similar in-universe term) is somehow activated (usually during puberty, but sometimes during a suitably traumatic origin event, or from birth) and grants the bearer superhuman abilities, and occasionally a dramatically changed (though not necessarily hideous -- but some are disgustingly ugly freaks) appearance. Marvel Comics specifically designates mutants as people who bear their mutation from birth (though, as mentioned, it may not express itself until later), distinguished from "mutates," who are genetically altered later.
On the other hand, in grimdark works, mutants are generally depicted as beast-like, malformed, monstrous, and otherwise sub-human creatures. They are often enemies to be destroyed without feeling guilty, or occasionally wretches to be pitied, especially if they started out "normal."
Warhammer 40,000
The Imperium of Man's thoughts on mutants can be summarized by the page quote: "Beware the alien, the heretic, and the mutant." After all, mutation of the body from the Holy Human Form is a sure sign of Chaos taint, and for that matter reflects poorly on the parents of the mutant in question, for surely the child would not be mutated if the parents were truly faithful to the Emperor!
That said, some mutants are tolerated in the Imperium, because they are necessary and/or relatively stable. This includes Navigators, who are necessary for Warp travel, psykers, who are important for astropathic communication, powering the Astronomican, and feeding the Emperor, and the various strains of abhumans, who are stable and useful.
On the opposite end of the spectrum, followers of Chaos treat mutants very highly, as they are considered favored by the Chaos Gods. Tzeentch in particular doles out mutations like there's no tomorrow (and knowing 40k, there legitimately may not be), giving his followers extra eyes, arms, legs, and even heads.
It's noted in the fluff that the reality is that most mutants are actually the result of legitimate flaws in the human genome, thanks to countless centuries of pollution, radiation, and bio-warfare; mutants caused by worship of the chaos gods make up a distinct minority in comparison. However, the Imperium treats all mutants the same, with a "better safe than sorry" mindset, and it's made pretty clear that this is actually why regular mutants are so eager to throw in with the forces of Chaos if given the chance.
Fallout
In Fallout, mutants are all over the place, both humans and animals. Fallout mutants are often associated with radiation, but the truth is that they are actually a result of infection from the bio-weapon Forced Evolution Virus, which escaped during World War III and interacted with the radiation to create the various monstrosities. The most iconic mutants in the Fallout series are the Ghouls, the Super Mutants, and the Centaurs.
Ghouls are humans who were exposed to both intense radiation and to F.E.V, which left them with burned, decayed, corpse-like bodies but biological immortality; the first ghouls were created during World War III, and have been around ever since. However, an increasing number of them are losing their minds and becoming mindless predators, "Feral Ghouls", and for this reason there is a lot of prejudice against Ghouls. The vast majority of the sane Ghouls are actually some of the nicest people in the games.
Super Mutants are humans who were quite genetically pure and so responded "well" to F.E.V. The result are hulking green humanoids, dramatically stronger and tougher than humans and biologically immortal, but also savage, violent, crazy and sterile monsters. There's also the Nightkin, blue-skinned and sneaky offshoots who've all gone nutters from overusing invisibility tech (usually multiple personalities) which is also what makes them sneaky despite being the same size as normal Super Mutants.
Centaurs are pretty much what happens to anyone else exposed to F.E.V; a mindless crawling mess of meat, kind of like a Chaos Spawn. Oh GOD BHHUBLUBBBBBBB.
As if that's not bad enough, the setting is home to a variety of genetically engineered horrors, such as the cazadores (giant versions of the parasitic wasps native to the Mojave Desert) and the night stalkers (fusions of coyote and rattlesnake with lethally poisonous bites, pack hunting behavior and the ability to swallow a human whole). Incidentally, both were created by the same government-endorsed scientist, who is not only still alive as a brain in a jar and robot body (along with his fellow scientists) but is still in the crater housing his DNA splicing lab. So there could be all sorts of new and horrifying creatures in the future.
The player can become one in Fallout 2 if he/she walks around for a bit in radioactive waste without proper foot protection (rubber boots). It's a sixth toe which can be amputated and then fed to people or eaten.
Judge Dredd
In Judge Dredd, mutants are generally treated as subhumans. They are the result of the ABC (that is, Atomic, Biological and Chemical) weapons deployed during World War III. They are banned from Mega-City One (though Dredd is working on changing that -- in fact, he happens to have mutant relatives, all of whom have a lantern jaw as big or bigger than his) and forced to live in the radioactive Cursed Earth.
Paranoia
In Paranoia, mutants are de-facto traitors. Friend Computer has as little tolerance for genetic deviations as the Imperium. Some mutants in Alpha Complex can register their mutations, avoiding mandatory execution. These mutants are required to wear an armband at all times identifying their status, and no matter how useful their powers this identifier will keep them from ever being really accepted in society. That is, if they survive long enough to fill out the registration forms without being killed first.
Mutant powers in Paranoia range from the typical and expected (telekinesis, mutant healing factor), to the hilarious and bizarre (Matter Eater). Nothing quite like walking away scott free because you ate the evidence before your arrest, especially when that evidence was a 14kg plasma rifle.
Of course, friend computer is perfect, and would never allow a mutant to be assigned to a Troubleshooter team. You can trust your teammates completely. This also means that there is absolutely no way that you have any mutations. Disregard what it says on your character sheet. The Computer is perfect, so your sheet must be in error.
There are rumors that there are some mutants who can telepathically communicate with machinery, such as robots, devices, and even Friend Computer itself. Rumors are Treason! But, if such a mutant did exist, it would be Extra Treasonous and would never be allowed to register its mutant power!