Undergraduate means simply 'below the level of a University degree'. Logically, this might be taken to include schoolchildren, and even those who have not yet gone to school; but in practice it is only used of those who have at least entered a university, but have not yet taken a degree. This is the greatest number of students in most universities in the UK.
Such students proceed in the natural course of events to a Bachelor's degree. After that, they are graduates (they have graduated) and either leave the university, normally to employment. If they remain, they are postgraduate students, working towards a Master's degree or a doctorate.
- You may want to see the article on graduate.