The study of the civilisations of Ancient Greece and Rome. As such it is a very broad discipline, encompassing language, literature, history, art, philosophy, linguistics and other aspects of both cultures.
The study of Classics is often dismissed as old-fashioned and irrelevant, but, to quote the Cambridge University Classics department website(http://www.classics.cam.ac.uk/):
In trying to see ancient experience for what it was, we discover again and again the hold which ancient categories, ideas, practices and institutions have over those which we have taken to be our own. Above all, the Greek and Roman worlds have provided modern culture with a tradition of ideals and values. They need our scrutiny and investigation. Perhaps more than ever.
Classics is generally similar to Classical Studies but, with Classics, the approach tends to be more based on reading texts in their original language
For a more detailed list of the topics encompassed in Classics, see the classical studies node