|30 kyu||29 kyu||. . .||2 kyu||1 kyu||1 dan||2 dan||. . .||5 dan||6 dan|
|<- weaker||stronger ->|
Kyu (pronounced "cue" as in pool cue) and Dan (pronounced like the nickname for Donald) are basically just two different Japanese words for "rank." The same system is used in martial arts like Karate -- a student starts at some high kyu (maybe with a "white belt"), and works his way down to 1 kyu (maybe "brown belt"), and is thence promoted to 1 dan, or "black belt."
It works out that this system is nearly transitive; if A gives B two stones, and B gives C four stones, A will be able to give C 6 stones for an even match. Essentially, the handicap system provides an informal empirical definition of the ranks. To find out what your rank is, you just play against another player who knows his rank, and see where you break even.
Players who come to go from other games tend not to want to use handicaps. In many game cultures, for example that of chess, there is a feeling that:
I've noticed that many chess players feel that if they have a good game, they might beat somone that's quite a bit stronger than them. This may be true in chess (for example, it's not so uncommon for an expert or a master to beat a grandmaster, perhaps in a speed game), but it's not true in go. It would be inconceivable for a 1 dan, for example, to beat a 6 dan. It just couldn't happen (short of a stroke in mid-game). Perhaps for that reason, and perhaps also because in go culture there's a great tradition of respect for stronger players, taking a handicap is not considered shameful in any way.
For one thing, professional ranks are not tied to playing strength in the same way that amateur ranks are. Professional ranks are conferred by these institutions according to certain rules, and as a result professional rank has as a lot to do with seniority. It's kind of like getting a Ph.D. As a rule of thumb, the strength difference between pro 1 dan and pro 9 dan is only about 2 stones. However, because of the competitiveness of the professional promotion system, many newly promoted 1 dans are very strong; sometimes stronger than some 9 dans.
In general, because all the institutions that grant diplomas make strength one of their main criteria, professionals are very strong. To a first approximation, all professionals much stronger than all amateurs. However, the strongest few amateurs are stronger than the weakest professionals because being professional is a matter of filling a formal requirement. Some talented amateurs who put as much effort into go as professionals do become as strong as a professional, but since they don't go through any professional promotion system, they remain amateurs. Going back to the higher education metaphor, an exceptionally smart and motivated non-academic might well know more about, say, computer science than a run-of-the-mill Ph.D.
The other side of this confusing coin is that there are many different strengths of amateur 6-dan. For example, I can handicap as an amateur 7-dan, but since there is no 7-dan rank, I call myself 6-dan. Meanwhile, there are amateur players who can give me one or two stones. Because there is no 7, 8, or 9-dan amateur, these players are also called 6-dan amateur.
The top professionals are frighteningly strong. A club of about 10 top players in Japan sort of trade the big-money titles among themselves from year to year. In Korea the title-holding club is only about 4 or 5. It's interesting, because the top players don't know more about go; the second-tier players are often as highly respected for questions of go theory as the title holders. But winning under tournament conditions takes a kind of mental toughness that very few people seem to have. It may be that a top player rarely plays a move that's significantly better than what a second-tier pro would have played if asked, but that under tournament conditions they just make fewer mistakes.