In the Go world, an amateur is who doesn't belong to one of professional organizations. Sometimes abbreviated as "am" or "ama". The words amateur and professional are antonyms.
Most amateur Go players play the game for fun, as a hobby, though some of them may derive significant income from go related activities, such as teaching, writing, publishing, organising or managing.
In the world of go, there has normally been a quite clear distinction between amateur and pro tournaments. Things have changed a little, quite recently, with some professional tournaments allowing amateurs to qualify via preliminary stages.
The Japanese tradition was that insei do not play in amateur tournaments; certainly not as part of their training. Amongst insei who do not become professional you can find some of the strongest 'amateur' players; so in this case a top amateur might be a student pro, who fails to pass the professional qualification tournament, or who decided to pursue another career or education, instead of becoming an actual pro. It used to be unusual for future pros to come from amateur go circles (Kageyama and Nakayama being notable exceptions). The great Japanese pro Takagawa Kaku, holder of the Honinbo title for nine consecutive terms, and a top player in the 1950's and '60's said when he retired that he was looking forward to becoming an amateur. An interesting example of movement from professional to amateur.
In China to be a professional a player must meet certain requirements for tournament play and teaching beyond just qualifying on the basis of strength. Consequently Chinese professionals may lose that status because they can or will no longer carry out the official duties.
On the other hand, in China, it used to be normal for the strongest amateurs to compete in the WAGC, before turning pro. In that way players such as Ma Xiaochun played as amateurs, on their way to becoming title holders. In China national tournaments for amateurs can give a chance to the winner to play as pro 1 dan, for some short period (as a candidate pro).
In South Korea promotion to pro is now regulated by a very tough league (with 80 players), only four becoming pro each year. The old system, described in First Kyu, changed many years ago; it gave a chance to 1 gup players to become pro. In South Korea there is now a genuinely Open tournament, in which any player in the world can (in principle) qualify.
The Fujitsu Cup, an international pro-am tournament, allowed a few top amateurs to qualify from around the world, getting games against pros. This has become more interesting, since upset victories are now not so rare.