The term "hybrid" does not refer to a singular race. Rather, "hybrid" is a classification of race, like "Magical" or "Monster".
Hybrids are conbinations of the other races, though most commonly of Non-Magical and Magical races. In most cases one race's traits will be the more dominant, so for classification, it's the dominant race that determins the [hybrid's][hybrids] classification.
Hybrids, on a whole, tend to be rare; there's no explicit reason for this, though there are strong societal pressures among some races to discourage this. However, Infernals, Celestials, and Dragons have a history of being fascinated with Non-Magical races, often disguising themselves as a member of those races and having liasons. The offspring that result from these liasons explain why most hybrids are mixtures with magical races.
Hybrids are, at the end of the day, their own races. In mixed parantage, one race will almost always be dominant. In the case of magical heritage, that race will always be dominant (though if there's a mixture of magical racial heritage, one or the other will be dominant, but never both).
True half-breeds —individuals whose parents are pure bloods of different races— will always be a hybrid, and if one of those races is magical, then it will be dominant. However, if the magical heritage is anything other than first generation, there's a chance it might assert itself, or remain dormant. However, when it does assert itself, the individual is just as powerful as a first generation hybrid, with all the same abilities, strengths and weaknesses that come with their hybrid race.
Hybrids have been around since before the sundering, when magical and non-magical races intermingled freely. Specifically, the Dragonborn were the first of the hyrbird races to be formed, a fact they are very proud of. Lilin and Nephilim were next, with Nephilim being highly admired and frequently worshiped.
After the sundering, hybrids became much more rare. Lilin in particular were reviled; it was believed they (and their Infernal ancestors) had something to do with it.
Feeling ill-at-ease in the current world, most hybrids either try to pass themsleves off regular members of their secondary race, or (as in the case of Dragonborn) attempt to remove themselves from the concerns of the world entirely.
Hybrids, for the most part, live in the Living Realm. The magical hybrids can survive in the magical realm without much issue, and some individuals have found thier way through portals and emmigrated, but it is not believed any large populations exist in the Magical Realm.