In the UK, university students are undergraduates between starting university and getting their first degree (BA for example). Studies after that point ( MA for example) are postgraduate studies.
'Graduate - or 'grad' - school' is an American term, and I can't recall ever having heard this used in a UK context. However, a quick google search informs me that a handful of UK universities have set up so-called 'graduate schools', presumably to make their function clear to an international candidature who are more familiar with American terminology.
So the answer to your question is the latter interpretation : they cater for graduates, in other words, students who have already been awarded their first degree. Many also co-ordinate doctorate studies.