The X-Men are the premiere mutants of Marvel Comics, but they're only the public face of the species known as Homo Superior. The mutant species is poised as the next step in evolution for humanoid life on Earth, and their mutations manifest in a variety of ways. While some mutants have rather mundane powers and abilities, some of their species boast gifts and power levels that are off the charts.
Omega-level mutants are the most powerful members of their species, with their levels being at the zenith of whatever their abilities are. This makes them both worthy allies and menacing enemies, depending on which side they fight on, and with most of mutantkind now joining Xavier's cause, the world just got a whole lot less threatening to the world's Omega-level mutants.
The idea of Omega-level mutants was introduced in 1986's Uncanny X-Men #208, though the concept wouldn't be explored further for some time. The most current definition for the term is that an Omega-level mutant is one who has reached the utmost power level of whatever their ability is and thus cannot be surpassed in it. For instance, Magneto has such a high degree of magnetic ability that no one on Earth could naturally surpass it... though his daughter Lorna, aka Polaris, may prove that untrue, soon enough.
In the case of Omega-level mutants with multiple powers, typically only one of these is ranked at an Omega classification. Such is the case with Jean Grey, who has both telekinetic and telepathic powers, but is only an Omega-level telepath.
Once the Omega-level classification was introduced, many of Marvel's mutants were retroactively given the status, with newer characters soon being introduced that were also Omega-level. These include classic characters like Iceman, Storm, Proteus, Legion, and the aforementioned Magneto and Jean Grey. Newer mutants like Quentin Quire and Mr. M were also revealed to be Omega-level, boasting incredibly psychic and matter manipulation abilities, respectively. Iceman and Storm have both shown phenomenal control of the elements in the past, explaining their status as Omegas, with Magneto and Jean doing the same.
The recent X of Swords event also introduced several new mutants who reside on the Arakko section of Krakoa. Led by Genesis, the wife of Apocalypse, all of these mutants are Omega-level, likely due to the harsher environments. Franklin Richards, the son of Reed Richards and Sue Storm was once considered an Omega-level mutant, but this was shown to be a manipulation on his part, using his powers to alter reality and "pass" as a mutant. Given how grandiose his reality-warping powers are, however, this begs the question of whether or not a true mutant with similar powers would ever be considered Omega-level.
Other mutant classifications include the obvious Alpha-level designation. Though it would seemingly be the opposite of Omega-level, Alpha-level mutants are actually incredibly powerful in their own right and include the likes of Cyclops. This explains why Mr. Sinister has been so interested in the powerful offspring of Cyclops and Jean Grey. The other known designations include Beta and Epsilon before finally reaching Omega-level.
A few mutants have been designated as beyond Omega-level, such as Gabriel Summers. Having once absorbed the mutant energy of others of his species, Gabriel, a.k.a. Vulcan, was described as being way past Omega-level. Franklin Richards was also once referred to as beyond this power level, but his mutant classification was, as mentioned, a ruse.
Other continuities and adaptations in different mediums would add their own classifications, though Omega-level is definitely the most well-known. Mutants with weak or arguably even useless abilities are not known to have a classification, and of course those born without the X-gene who later develop powers are "mutates." With the nation of Krakoa seeing mutants on a united front, there's a higher chance than ever that more Omega-level mutants will soon walk the Earth.