That's a very good question! It really depends where you are, and who you ask.
Technically, a "postgraduate student" is one who has a Bachelor's degree and is studying a non-degree programme, like a certificate or diploma (not a Masters or PhD). A graduate student is one who is enrolled in a Masters or PhD programme.
However, the terms are used very interchangeably these days. "Postgraduate student" is more common in BrEng and "Graduate student" is more common in AmEng.
We can also say "grad" and "postgrad", but those terms are quite informal.