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, unless you go to a species like bacterias or viruses that move though few thousands or even billions of generations per year (as opposed to human 0.05 per year), or Cephalopods which evolve via RNA editing, which can sometimes evolve without needing natural selection.
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, due to the implications of such genes, mutants are classified as a separate sub-species of humanity despite the fact that it essentially rapes basic biology up the ass and gives evolution the middle finger, alas this is what most would call as comic bullshitium. 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."
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; in Imperial space 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.
An exception is found by Eisenhorn when he tracks a REDACTED TO PREVENT SPOILERS to the agri-world of Eechan; the world has a large lower-caste population of mutants who work the farms and crop-mills. These mutants are tolerated because purging them would mean the loss of food supply to many systems until the world can be repopulated, and it is possible they are mutant because of a factor caused by living on Eechan itself.
However, the state on which mutation becomes a crime varies between worlds. Usually, minor mutations such as bizarre skin color or an extra finger or toe is often brushed off or easily fixed via surgery. Major mutations such as an extra set of eyes or reptilian or slimy skin would be more likely to be looked into by either the Arbites or the Inquisition as the chance of these being related to Chaos is unsurprisingly quite high. Because of this, some of these mutants often flee into the deepest pits in Hive Worlds or escape and hide within the vastness of space. Usually these mutants find companionship and trust for the Lost and the Damned whom they use as specialized shock troops due to the fact that no two mutants are the same and have differing qualities and quarks that can prove advantageous in battle. These rag tag team of mutants are called mutant rabbles and are often undisciplined and rowdy.
Usually the laws of Imperial worlds normally forbid such creatures from possessing armaments; those weapons that mutants do possess are usually limited to crude shotguns or blunderbusses, heavy duty revolver-stubguns, chains, whips and clubs which can be easily made and concealed.
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 known as the Forced Evolutionary Virus (F.E.V. for short), 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, dim, and sterile monsters. Oddly, there are cases where Super Mutants don't undergo a mental change, maintaining human intelligence and can even be quite civilized, even nice. 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.
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.
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. Eating the evidence is Treason. Treason is punishable by Death.
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. Sheets in error is Treason. Treason is punishable by Death.
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! Treason is punishable by Death.