Apocalypse is a sort of meta-mutant. His original mutation seems to be shapeshifting, but he was also an X-ternal (the Eternals are a race of immortals; the Celestials took proto-humanity, about a million years before, and changed them into three groups: the Deviants, genetically unstable people who often look monstrous, but have an advanced society and dominated humanity for a long time until they were driven underground; the Eternals, who are immortal and possess god-like powers, but who are very few in number and live in the high places of the world; and humans, who seem unassuming, but have the potential to develop superpowers to surpass even the Eternals. We call such people 'mutants'. Some of the mutants are immortal, and are called X-ternals).
He discovered an alien ship buried on Earth and claimed it for his own; it changed him into something which was not just a mutant anymore, or even just an X-ternal.
Apocalypse's power is to change into pretty much anything he wants, organic or cybernetic; it's complete control over his body's molecules. Apparently he can also increase his mass, like Giant-Man or the Hulk. He can also transfer his essence into new bodies; in the distant future, approximately the 40th century, he has become an old, wizened shell who has to change bodies more and more often, because he's burning them out. That might be where the image of Apocalypse in an armor comes from:
This is what he originally looked like, in his misspent youth, 5000 years ago:
Interestingly, En Sabah Nur is currently once again incarnated - proving he cannot be permanently killed; he's just reborn - but as a boy, Evan, whose sheltered upbringing actually makes him pretty much the opposite of what Apocalypse is.