This post will give a brief overview of the 'universals debate' in metaphysics, and my thoughts on it.
A 'universal' is special sort of entity, one which can be possessed or exemplified by many different objects ('particulars') simultaneously. The idea is that when different objects agree in attribute, this is because there is some part of them which is literally shared by both: namely, that they both exemplify the same universal. For example, when we say "Socrates and Plato are both wise", we are saying that they both possess the same special entity (universal) of 'wisdom'.
At least, that's what metaphysical realists suggest. Nominalists, however, deny that (mind- and language-independent) universals exist. They say there is not actually any such entity as 'wisdom' (for example) floating around in the world. To say 'Socrates is wise' is taken to be a basic unanalysable fact about the particular thing that is Socrates (rather than interpreting it to mean 'Socrates possesses the universal of wisdom', which the realist takes as being the basic unanalysable fact).
I see realism here as a sort of last-resort: we really want to be able to explain the world without postulating any unnecessary entities (ya know, Ockham's Razor and all that). The question then, is whether universals are necessary...
The challenge for Nominalism:
The biggest problem for nominalism is to explain our use of abstract reference (eg 'Wisdom is a virtue', or 'Red is my favourite colour'), i.e how it is that we can speak truly about universals if they don't exist. The usual method of response is to somehow translate sentences which seem to be about universals, to instead be about something else (and so avoiding any mention of universals. Adjectives are okay, since they merely describe particulars. What we need to avoid are those dratted abstract nouns!). The big question is what that 'something else' should be.
Austere nominalism suggests the answer is particular objects (eg 'wisdom is a virtue' gets translated as something like 'wise things are virtuous things'). Such translations can run into difficulties however (though I won't go into details here), so this seems a less than ideal option.
Metalinguistic nominalism recommends interpreting talk about universals as instead being talk about language. Here, 'wisdom is a virtue' becomes 'the word "wise" is a virtue predicate', and 'courage is a property' becomes 'the word "couragous" is an adjective', etc. This can eventually form a coherent (and not too complicated) theory, but it's so counter-intuitive I find it difficult not to discard the whole approach as just plain wrong. After all, we generally think that when we're talking about wisdom, we're really talking about the concept of wisdom, rather than something so trivial as the mere word.
So that suggests an alternative answer... that talk about universals is really talk about concepts.
I'm kind of making this section up as I go along - there was nothing on it in the textbook, and our lecturer only mentioned the possibility in passing, so I don't know what the details are supposed to be like. So I'm making up my own details. Here goes...
It might be thought that concepts are universals, that this position is no different from realism. However, recall that the realist's universals are supposed to be mind- and language-independent; entities which really exist out there in the world somewhere (though possibly outside of space & time). Concepts, by contrast, are mind-dependent; they only exist in people's minds.
Apart from that crucial difference, I do see concepts as being fairly similar to how a realist conceives of universals. A few key points though:
1) Predication: Objects may 'exemplify' concepts, but they do so in the sense of 'being a good example of', rather than the 'possession' associated with universals. That is, an object exemplifies a concept if it shows relevant similarities to the paradigmatic instance of a concept. For example, we have a concept of 'wisdom', and a rough idea of what it means for someone to be wise. So when we say that 'Socrates is wise', we are saying that Socrates is relevantly similar to this paradigmatic case we have in mind (i.e. Socrates behaves in a relevantly similar way to how our idealised conception of a 'wise person' behaves).
2) Attribute agreement: Similarly, to say that 2 objects agree in attribute is not to say that they share some metaphysical essence, but rather, it is because we see them both as exemplifying (being a good example of) a single concept we have in mind.
3) Abstract reference (i.e. abstract nouns like 'wisdom') is used not to refer to universals out in the world, but rather, concepts within our own minds. To say that 'wisdom is a virtue' is to say "I have this concept of wisdom, which exemplifies ('is a good example of') this other concept of virtue".
That translation only sounds complicated because I also had to include the explanation of predication (for the general 'X is a virtue'; see #1 above).
For a simpler example: 'Bob prefers red to blue' becomes 'Bob prefers the concept of red to the concept of blue'. It's an extremely simple procedure - whenever you come across an apparent reference to a universal, simply replace it with a reference to the appropriate particular concept.
So... that's my initial (rough) thoughts, anyway. Do let me know if you see any flaws, or have any suggestions, etc.