Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that is wholly non-spatial and non-temporal (i.e., that doesn't exist in space or time) and, hence, is entirely non-physical and non-mental. This might be a bit obscure, but in what follows, we will go into much more detail on this and see precisely what the platonist view amounts to. Before we do this, however, it should be noted that the view we will be discussing is a contemporary view. Now, there is no question that this view owes a great deal to the writings of Plato — that, of course, is why it's called ‘platonism’ — but it is not entirely clear that Plato actually endorsed this view, and it is for this reason that the term ‘platonism’ is spelled with a lower-case ‘p’. It may be that Plato endorsed platonism, and indeed, it can probably be said that this is the standard view among Plato scholars, but the question is still controversial.
More here: http://plato.stanford.edu/entries/platonism/