What Is a Professional Go Player?
A professional Go player in Asia is a person who earned this status from one of the professional Go associations in Japan, China or Korea.
A professional can make a living based on Go. Only a few top professionals can live just from the tournament prizes - the vast majority earn their living by teaching amateur players for money.
There are a few hundred professional Go players in Japan - but several tens of millions of amateur players. There is about the same ratio in China and Korea as well.
Here is a video from one of the international professional Go tournaments:
How to become a professional Go player?
In order to become a professional Go player in Japan, one first has to become an insei and to be in the top few of the annual insei competition. The exact number of new professionals every year is decided by the Japanese Go Association - normally it is around 5 or so.
There is also one place for the public in the final tournament (which is an 18 person round-robin), so theoretically one doesn't need to be insei. There is a wonderful book called "First Kyu" which describes the incredible struggle young persons go through in order to try to become professional Go players, from the perspective of Korea in early 70s. Some things are different today, of course, but many of the aspects described in this book are eternal truths.
There is also a special annual competition for the insei girls. Professionals qualified this way have a more limited pro status until they reach a particular level (I think that is 3 dan).
There are special rules for non-Asian insei: even if they don't qualify the normal way, just by having consistent good results they can become professionals in Japan. (Not sure what "consistent good results" mean nowadays - as far as I remember it used to be qualifying 3 years in a row in the final tournament). This "handicap rule" is meant to promote Go among non-Asian players. And the same limited pro status applies as for the girls tournament.