Postgraduate simply means 'after graduating', or 'after having been awarded a university degree. In practice, like undergraduate, it is rarely used outside a university context: doctors and lawyers rarely describe each other as 'postgraduate', although all members of these professions are, by definition, postgraduate.
In universities, the term is mostly applied to students - those who have already been awarded their Bachelor's degree, but are now working towards one of the higher degrees, the Master's degree or a doctorate. By extension, postgraduate demonstrators, etc, are junior teachers in universities drawn from among the postgraduate students who have not yet completed their second or later degree studies. (Logically, all university teachers can be described as postgraduate. In practice, only those who have not fully qualified are called postgraduate.)
- In the USA, what are labelled here 'postgraduate students' (etc) are often called graduate students. This is logical: such people, working towards a second degree, have already been awarded their first degrees, hence are graduates.