X-Men
Mutants
Human Mutants have many distinctive traits in regards to biology, physiology and genetics from other humans. Mutants can have wildly varying biologies depending on the nature of their mutation. Mutation is caused by the X-Gene in the mutants' DNA. Mutations normally manifest themselves during puberty, often as a result of a traumatic event. Some mutants exhibit their mutations at birth, however, while others do not manifest their powers until adulthood. Secondary Mutation is a phenomenon in which an existing mutant gains additional powers, or a change in their appearance due to a second mutation. The mutants can be divided into multiple categories, regarding to their origins, genetic status, powers (e.g. Omega Level Mutant), etc.
While in most Marvel universes the X-Gene was presumably a result of modifications on early man by the Celestials, in the Ultimate Universe the X-Gene is a creation of the US government weapons project in Canadian facilities. This reality's Wolverine was the first of these mutants, and was called the Mutant 0. This mutation is similar to a virus, infecting the population slowly and turning them into mutants.
X-Men
The X-Men are a team created by Professor Charles Xavier as a response to the emergence of naturally occurring super-humans known as mutants among the baseline human society. Seeking to promote his dream of peaceful coexistence between mutants and humans, Xavier trained his X-Men in the use of their powers at the Xavier's School for Gifted Youngsters, training them to become defenders of a world that hates and fears them. Over the years, the heroes of mutantkind found their mission constantly endangered by both reactionary humans and fanatic mutants.
Antagonists
Mutants in general, and the X-Men in particular, face a different type of enemy than other metahumans. First of all, the primary enemy of the X-Men is not a greedy plunderer, or an insane tyrant, or an omnicidal alien. The true enemy of the X-Men is Hatred. Hate and Fear of the unknown, the different.. the mutant. Protecting mutants from persecution and misunderstandings, and protecting humanity from the depredations of Evil Mutants, the X-Men stand united against this one pervasive adversary.
More interestingly, the X-Men and most of their rivals tend to share cultural and genetic origins. The Brotherhood of Mutants, the Mutant Liberation Front, and several other people whom the X-Men have clashed over for noble reasons over time, have time and time again sided with the X-Men when the threat came for their race as a whole. Sentinels. Friends of Humanity. The Phalanx. The one unifying tread all mutants have, their need to survive past being the subjects of hate or convenient scapegoating.
As such, very few 'Evil Mutants' actually exist that have not, at one point, sided with the X-Men against their shared common enemy. This is why over time, practically everyone of the X-Men's enemies have worn an X somewhere on their costume, or have been otherwise associated to the Xavier Institute.