If you do something full-time, you spend most of your available time doing it. Working full-time usually means spending around forty hours a week doing your work.
You can be employed at a full-time job, or you can be a full-time student. Some people are full-time parents, and you might hope to be a full-time artist or a full-time musician one day. In all of these examples, people use the majority of their waking hours (at least during the work week) focusing on one thing. This term dates from the late 1800s.