How do ranks work in amateur Go?
Let's start with the basics. Beginners start out as "30 kyu". As they get stronger, the number decreases to 29 kyu, 28 kyu ... down to 1 kyu. Kyu means "student" and is commonly abbreviated 'k'. The range from 10k to 30k is called DDK (double digit kyu) while 1k - 9k are SDK (single digit kyu).

The difference between each rank is supposed to be 1 stone of handicap. That is, an 8k should give 3 handicap stones to an 11k. All handicap games are played with komi of half a point. So an 4k playing a 3k plays black, not placing any additional handicap stones, and white receiving 0.5 komi.

Once a player surpasses 1k he becomes 1 dan, "master", abbreviated 'd'. Dan ranks go from 1 dan ('shodan') to 7 dan for amateurs (occasionally to 8 dan).

Professional players are obviously stronger than amateurs, since they studied the game very intensely for years and received training by pros. Only top amateurs have a chance of winning. Pro ranks go from 1 dan pro to 9 dan pro (abbreviated 'p' instead of 'd') with "10 dan" being a title that can be won in a tournament.

Nowadays, the difference in strength between pros is not related to their rank, because new 1 dan pros received the most up-to-date training and already possess vast knowledge (just like someone coming fresh from university is very up to date on the current research). Formerly, it was said the difference between 3 or 4 pro ranks is about 1 stone.

That being said, there are different ranking systems in different parts of the world that are not completely equal for various reasons. Online go serves got their own ranks as well. On Go servers, it is usually possible for amateurs to get assigned a rank of 9 dan.