When the great Stan Lee was developing the concept for Marvel Comics' X-Men with Jack Kirby, it was decided that they would create a superhero team without the need to explain away the characters' powers as being the result of science gone wrong, as he did when creating Spider-Man with Steve Ditko or Hulk and the Fantastic Four with Jack. Thus, the idea of Marvel "mutants" was conceived to describe humans who were simply born with extraordinary abilities, inspired by real mutations that exist in nature.
Since then, the Marvel Universe's mutant population has grown exponentially, and with that growth came the escalation of power possessed by these individuals. The term "Omega-level mutant" was eventually established to classify mutants of immeasurable potential within their specific category of ability. Several such beings (or ones close to them) have appeared throughout the history of X-Men and Marvel to astound us with what they are capable of.
En Sabah Nur
One of the first mutants in existence, En Sabah Nur, better known as Apocalypse, was raised under the dogma of survival of the fittest, which he would seek to impose upon humanity, orchestrating events to weed out the weak and allow only the strong to flourish, among whom he counted himself, and not without good reason.
Effectively immortal, Apocalypse has evolved and adapted to acquire a myriad of abilities over his millennia-long life. Originally a being of incredible strength, speed, and endurance, he enhanced himself further using alien technology, becoming capable of energy manipulation, psionic powers, and reshaping his molecular structure to form weapons, self-heal, and change size. While not officially listed as Omega-level, he remains one of the most powerful mutants to ever walk the Earth.
Erik Lehnsherr
Born Max Eisenhardt and later recognized as the mutant terrorist Magneto, Erik Lehnsherr was a Holocaust survivor who desired to build a world dominated by his fellow mutants, deeply influenced by his experiences and personal tragedies. Dubbed the "Master of Magnetism", Magneto's mutation enables him to create and manipulate magnetic fields to accomplish a range of feats, such as moving ferrous metallic objects of staggering masses, erecting nigh-impenetrable protective barriers, and self-sustained flight.
However, Magneto's power actually extends beyond magnetism to encompass all aspects of electromagnetism, one of the four fundamental forces of nature, allowing him to rearrange matter by affecting atomic bonds, manipulate electricity, light, and all other forms of electromagnetic energy and radiation, and even control the polarity of the Earth's geomagnetic field.
Jean Grey
A member of the original team of X-Men, Jean Grey has gone from sweet-natured powerhouse to planet-destroying villainess to perpetual Lazarus figure. She was initially portrayed as only having telekinesis, the power to control matter with her mind, but was later revealed to also be a telepath, someone who could read and influence minds. The latter ability was temporarily suppressed by her mentor, Charles Xavier, ostensibly because they were too much for her to control though that line of thought has been extensively questioned over the years.
In both her telekinesis and telepathy, Jean has demonstrated remarkable talent, achieving such effects as exciting molecules to generate friction and altering others' actions and perceptions. Her considerable psionic capacity made her a favored host for the cosmic Phoenix Force, which could boost Jean's powers to where she is able to affect matter on the subatomic level, project vast amounts of energy, and resurrect herself repeatedly.
Kevin MacTaggert
The mutant son of geneticist and longtime X-Men associate Moira MacTaggert, Kevin MacTaggert, or Proteus, as he is famously called, possesses dangerous reality-warping powers that required him to be confined within his mother's research facility before he escaped to wreak havoc upon her and the X-Men.
Proteus can transmute matter and energy into any form or state he wishes, twist space around him, and essentially defy all manner of physics and logic. Unfortunately for him, his powers are so potent that they exhaust and destroy his physical body, forcing him to take possession of others, which he is able to do due to being composed of pure energy. Aside from that, his only other weakness is inorganic materials, specifically metal, which his energy form cannot tolerate.
David Haller
X-Men founder Charles Xavier is the world's foremost mutant psychic, so it stands to reason that his offspring would be one of comparable might, which David Haller, a.k.a. Legion, certainly is, and then some.
David suffers from dissociative identity disorder, with each of his alternate personalities possessing some kind of superpower. His first known alter was telepathic, like Xavier, but he has since revealed countless more personas with abilities like telekinesis, pyrokinesis, reality and time manipulation, and various others. David's mind can also manifest additional empowered identities in reaction to outer or inner stimuli, further contributing to his already tremendous power set.
Gabriel Summers
The long-lost younger brother of X-Men members Scott and Alex Summers, Gabriel Summers, codenamed Vulcan, through a series of harrowing circumstances, grew into a bloodthirsty intergalactic imperialist with a grudge against the X-Men.
The Summers brothers have in common that they are living batteries of energy, but Vulcan leaves both his older siblings in the dust with respect to raw power. He is capable of absorbing immense volumes of practically every known type of energy (heat, light, electricity, radiation, etc.), including more exotic forms like Scott's optic beams and magic, and releasing it in devastating blasts or molding it into different shapes. Vulcan can also bend existing energy sources to his will as well as use energy to fly at blinding speeds and survive under the harshest conditions.
Nate Grey
In an alternate timeline where the mutant Darwinist Apocalypse has conquered much of the world, his treacherous lieutenant Nathaniel Essex/Sinister engineered the ultimate weapon to one day take out his master in the form of Nate Grey, the genetic progeny of Jean Grey and Scott Summers.
At his peak, Nate could harness the virtually limitless psychic energies of the Astral Plane to become possibly the most powerful psionic mutant who ever lived. His abilities included being able to connect his mind with numerous others simultaneously and over great distances, give physical form to astral entities, and perceive the future. Nate's power and skill as a telekinetic was believed to be unequaled, capable of leveling entire continents or performing feats that require extremely fine control such as shifting the atoms of solid obstructions around his body so that he passed through them harmlessly.
Hope Summers
Following the conclusion of the "House of M" storyline, which significantly reduced the planet's mutant populace as a result of the actions of the Scarlet Witch, Hope Summers was the first new mutant birth and a prophesied messianic figure meant to save mutantkind and humanity.
Hope has the ability to mimic and manipulate the powers of other mutants. Any power that she copies, Hope can utilize and command it at its highest capacity, perhaps even to a greater degree than the originator of the power, and she is able to hold onto and wield multiple abilities at a time. Furthermore, Hope can regulate, augment, and jumpstart powers in mutants, granting them easier control over their gifts or awakening their true strength.
Franklin Richards
They say that children are born with boundless potential, and in the case of Franklin Richards, son of The Fantastic Four's Reed Richards and Sue Storm, that expression quite fittingly applies.
From infancy, Franklin displayed astonishing measures of ability that made him a target for many supervillains and formidable entities who sought to control it. This power allowed Franklin to influence, direct, and reconstruct the fabric of reality and its constituent elements at magnitudes that were deemed god-like. He could create whole new universes, overpower cosmic beings, and turn imagination into fact. Franklin's mutant status may be questionable now, yet his reputation as one rises above it all.
Absolon Mercator
Best known simply as Mister M, Absolon Zebardyn Mercator is just as much an enigma as he is seemingly all-powerful. Gifted with the power to transfigure matter and energy, down to their atomic and subatomic structure, Mister M can do... presumably anything.
His proven capabilities have varied from repairing or dismantling complex machinery to stimulating and accelerating biological processes like cellular regeneration to conjuring elemental forces. Mister M's exact limits, if any, are unknown, meaning that he could very well be omnipotent.