Briefly, the argument might be put as follows. If a man is made to be what he is by participating in a Platonic Form (though Greek did not distinguish small and capital letters), then another Form will be needed to explain how both the man and the Form can be called 'man'.
This Form will be a 'third man', and yet another Form (a 'fourth man') will be needed to explain how these three items can all be called 'man', and so on to an infinite regress.
As Plato presents it, the argument requires Forms themselves to have the properties they are Forms of (see paradigmatism), but the argument can be restated to avoid this, and in fact raises fundamental questions about objects and their properties, and predication.
The theory has been copiously discussed in the 20th century, and by Aristotle (384-322 BC) - to whom we owe the name, and also the example 'man'; Plato's own example being 'large' - for whom it played a major part in determining his reactions to Plato's metaphysics.
J A Passmore, Philosophical Reasoning (1961), ch. 2; modern treatment of Plato's problem