When we say "Rex is a dog", or "Rex is brown", we have a subject/predicate form of sentence. With the word "Rex", we are referring to an individual dog. But what about the predicate? Does it refer to something as well? If it doesn't, then isn't the word just meaningless gibberish? If it does refer to something, what is it and what is it like?
"Rex" we would call a particular, or an individual. He is located in space and time, and you can point to him. "Brown", or "dog", we would call a universal. They are abstract and not located in space or time. The problem of universals is the problem of how to account for what the predicate is, what it refers to, and what it is like.
II. Solution: Realism
Realists say that universals really exist, as abstract objects. There is such a thing as "dogness" and "brownness". Individuals then participate in those universals. Plato thought that universals existed in a third spiritual realm, neither mind nor matter, and that the physical world is merely an imperfect instance of that perfect realm of universals, or what he called Forms. Plato's version of realism is vulnerable to the Third Man Argument: to be the kinds of Forms they are, they in turn must participate in yet another form higher up, and so on to infinity. Aristotle agreed that universals are real, but are only located in the individuals themselves.
III. Solution: Nominalism
To get rid of extraneous entities, nominalists want to say that universals do not exist at all in any form. There are just individuals. There is Rex, Spuds, etc., but no "dogness" or "brownness". One version, called resemblance nominalism, says that things only resemble each other. But then universals creep back in, because they participate in the "resemblance" universal, which is their predicate. Another version called trope theory states that everything consists of bundles of individuals, so "Rex" consists of a bundle of such individuals as Rex, dog, brown, etc.
IV. Solution: Conceptualism
Similar to nominalism in that it denies that universals exist, conceptualism states that universals exist solely in our minds.
For a more in depth look at the problem of universals, see the full article.