A student is a person who is studying at a school, college, university, etc.
An undergraduate is a student who is studying for their first degree at a university or college.
In BrE , a graduate is a person who has completed a first degree at a university or college. In NAmE graduate is usually used with another noun and can also apply to a person who has finished high school.
a high school graduate
a graduate student
A postgraduate is a person who has finished a first degree and is doing advanced study or research. This is the usual term in BrE , but it is formal in NAmE and graduate student is usually used instead.