When Stan Lee and Jack Kirby first gave us the X-Men, they simply imagined that there would be a few rare humans born with a special mutation that gave them an extra power or quality that set them apart.
As issues went on, we learned there were many more mutants out there, so much so that many humans feared being replaced or conquered.
The world of the X-Men has evolved quite a bit over the last several decades. We've learned that each mutant, along with having unique brain wave patterns, is born with a mysterious X-gene that causes their abilities, usually activating during puberty. And from the beginning, we've seen that not all mutants are cut from the same cloth. Some look just like us, whereas others could pass for aliens, demons or animals.
Over time, a system of classification came about, one to simply help the X-Men, along with agencies such as S.H.I.E.L.D., keep track of who had what basic traits and power levels. Of course, these aren't set in stone, since some mutants later tap into unseen power, are genetically altered by outside forces, or have experienced secondary mutations that change their nature and, as a result, put them into a different classification.
So for your benefit, here are the six basic categories of mutants in the Marvel Comics universe.
An Alpha level mutant is someone who looks no different from any other human being, and whose powers are pretty effective in combat. Likewise, they have no limitation placed on them as a result of their X-gene.
Kitty Pryde looks like any normal lady and is capable of walking through walls. Dazzler seems no different than others but can absorb sound waves and redirect them as light or lasers. Colossus looks like a living metal man when he uses his power, but he can then turn it off and resume the appearance of a normal person.
The mutant Rogue is now an Alpha level mutant since she can control her power, but once was a Beta level.
Beta level mutants are just like Alphas, but with a catch. While at first glance they can pass for a human being, careful scrutiny reveals they have some strange trait that sets them apart, like an eye color that doesn't generally appear in humans. A Beta mutant can also be someone who passes for human but has some limitation imposed on them because of their power, such as not being able to make skin-to-skin contact (which describes Rogue's condition for many years).
Cyclops is a Beta level since he can't turn off his power and must always wear special lenses or a visor to control or contain it. Gambit can control his power and mostly seems human, but he has red eyes and black scleras. Wolverine may seem human at first, but along with not being able to turn off his powers, he has an extra set of bones and muscles in each forearm, thus his famous retractable claws.
Like Alphas and Betas, Gamma level mutants are pretty handy in combat situations. But as a result of their X-gene, they cannot pass for human without the use of a disguise of some sort.
Blink's power of creating teleportation discs and energy projection makes her pretty handy in a fight. But along with this, she has lilac skin, eyes with no visible pupil or iris, pink hair, and strange facial markings. Nightcrawler can cling to surfaces like Spider-Man and can teleport to places he's seen (within a limited radius), but was born looking like a demon, complete with indigo fur, only three digits on each hand, two major digits on each foot, blank golden eyes, elf-like ears, fangs and a prehensile tail.
Delta Level mutants are those who, like Alphas and Betas, can pass for human at first glance. But unlike Alphas or Betas, their power is not something that is obvious and automatically combative. Their abilities are more subtle, enough so that an outside observer can't actually see the power in use. Delta mutants may go for years not realizing that they are, in fact, mutants, due to the nature of their abilities.
Domino is a Delta mutant. When she finds her adrenaline rushing, she causes probability to win out in her favor within her line of sight. But even this is not automatic, as she needs to actively take advantage of the "good luck" she causes. If Domino is being shot at, she can't just stand there and count on the bullets missing her. But if she starts running from those shooting at her, she'll find that most of the time she'll wind up narrowly avoiding each shot. Another Delta mutant would be Forge, who can instinctively build and rework machines without being told or even fully understanding how they work.
Epsilon level mutants either have a non-combative power or no real powers at all, but are like Gamma level mutants in that they cannot hide what they are. Many examples of such mutants can be found among the Morlocks, a community that choose to live in the sewers rather than among humanity.
Tommy of the Morlocks had strange colored markings on her face and could become paper-flat at will. Artie Maddicks' X-gene caused his body to alter, turning him into a pink, hairless being with growths on his head and large, blank eyes. Along with this, he lost his powers of speech and could now only communicate by projecting holographs of his thoughts.
These are the people who truly make normal human beings and government agencies afraid. These are mutants whose power levels are such that they are considered the most serious threats, capable of wide-scale destruction and proving very difficult to take down, contain or kill.
An obvious Omega level mutant is Magneto, whose powers over magnetism lead to a variety of abilities and also allow him to manipulate related fields of gravity and electricity to a certain degree. Storm could be considered an Omega level mutant, with her manipulation of the elements (an ability she can't completely turn off, as her power is keyed to her emotions). Iceman is proving to be another example of an Omega level mutant; though initially seeming more like an Alpha, Iceman has finally begun tapping into his full potential in the past few years, most recently showing he could create copies of himself to act as an army of organic ice beings.