As the others are saying, the Omegaverse is basically a collection of tropes about a world in which people's external and internal sex organs differ^[Rather, external and internal sex organs differ at a significantly higher rate and in a significantly different way than in real life. For real-life intersex variants in which external and internal sex organs differ, see Persistent Müllerian Duct Syndrome (PMDS), in which a person with a normative penis at birth has an underdeveloped uterus; and Complete Androgen Insensitivity Syndrome (CAIS), in which a person with a normative vulva at birth has atrophied testes in place of ovaries.], which in practice means you have "alpha", "beta", and "omega" variants of each sex, and which variant you are becomes known to you at puberty. For a summary of gametes:
| | Prader 6 (normative penis) at birth | Prader 0 (normative vulva) at birth | |
|
|
| | Alpha | ᯡ | 0 + ᯡ | | Beta | ᯡ | 0 | | Omega | ᯡ + 0 | 0 |
Although note that the types listed here as capable of both impregnating and being pregnant are in some Omegaverse stories only capable of being pregnant, and that even if you pair a type capable of impregnating with a type capable of being pregnant, that many stories also distinguish who may be impregnated by whom. Yes, that's 6 × 6 = 36 potential pairings for you to remember the dynamics of, which may vary considerably between stories! In any case, betas are usually presented as basically the "boring average people" variant encompassing the majority of people, whereas alphas and omegas are presented as Naturally Dominant™ and Naturally Submissive™ respectively, and Destined for each other. Reproductive heat also plays a big role in the Omegaverse, with heat suppressant drugs that always seem to fail at the most inopportune time due to overpowering alpha pheromones or whatever. You might also have stuff like penile knots on the alphas, and alphas might bite the napes of omegas to "mark" them as their own. So all in all, the Omegaverse is basically like werewolf smut meets mpreg. It's never been my thing — this description is just cultural osmosis with a bit of double-checking Wikipedia etc — but even with all of the Omegaverse's gender-related baggage, I have to respect it, because it clearly means a lot to a lot of people if so much collective fan labor has gone into the worldbuilding; and given the sorts of things I've read on AO3, I'm not exactly fit to cast the first stone here.
What's more interesting to me is how the Omegaverse basically started with somebody writing real-person fanfiction smut about the like lead actors of Supernatural I think, and then it just snowballed from there until now there's an actual, decidedly non-pornographic TV anime about the Omegaverse.
You should watch SPARKLE ON RAVEN: The Life of DrillGirl; it is a very good show. {Cēterum|Furthermore} {cēnseō|I think} {iūs auctōris|copyright} {esse dēlendum|must be destroyed}.