An undergraduate degree is the first degree you take, ususlly taken by people between the ages of 18 and 22. Most such degrees are B.A. (Bachelor of Arts) or B.Sc. (Bachelor of Science).
A graduate degree (or postgraduate) degree is one taken by somebody who already has an undergraduate degree. These include M.A. (Master of Arts), M.Sc. (master of Science) and Ph.D. (Doctor of Philosophy). A postgraduate degree is a higher qualification than a first degree.
just to clarify:
undergraduate degrees: BA, BSc
postgraduate degrees: MA, MSc
Doctorates: PhD
NB. technically PhD is a doctorate not a postgraduate degree.