Universals (used as a noun) are either properties, relations, or types, but not classes. It is worth noting that all four items are generally considered abstract, nonphysical entities. They are at least so considered by Platonic realists; there are others who use the terminology of properties, relations, etc., but who do not wish to be realists. Part of the difficulty, indeed, of understanding this problem is understanding the complex and confusing relations between theory and language, and what the use of language does, or does not, imply.
Consider some examples of universals: there are types, like dog or "doghood"; properties, like red or redness; and relations, like betweenness or "being between"; those are all universals. Any particular dog, particular red thing, or particular object that is between other objects is not a universal, but a particular, and instances of universals (or objects that somehow bear universals). Doghood, redness, and betweenness are common to many different things. So a universal is something that can have instances; but it does not make sense to talk about an instance of a particular.
Realists invite us to think of universals as the referents of general terms. In other words, they are what we refer to, when we use general words like "doghood," "redness," and "betweenness." By contrast, we refer to particulars by using proper names, like "Fido," or definite descriptions that pick out just one thing, like "that apple on the table."