Achievement in Judo is recognized by a series of ranks. The student ranks are called kyu and are usually differentiated by colored belts (obi). Different colors may be used around the world and in some countries there are more than 6 kyu ranks. The ten black belt, or expert, ranks are called dan. The traditional Judo ranks are:
English Japanese 6th grade rokyu 5th grade gokyu 4th grade yonkyu 3rd grade sankyu 2nd grade nikyu 1st grade ikkyu 1st degree shodan 2nd degree nidan 3rd degree sandan 4th degree yodan 5th degree godan 6th degree rokudan 7th degree shichidan 8th degree hachidan 9th degree kudan 10th degree judan
In the days before Kano created Judo, there was no kyu/dan ranking system in the martial arts. A more traditional method of recognizing achievement was the presentation of certificates or scrolls, often with the secrets of the school inscribed. Kano started the modern rank system when he awarded shodan to two of his senior students (Shiro Saigo and Tsunejiro Tomita) in 1883. Even then, there was no external differentiation between yudansha (black belt ranks) and mudansha (those who hadn't yet attained black belt ranking).
Kano apparently began the custom of having his yudansha wear black obi (belts) in 1886. These obi weren't the belts karateka and judoka wear today — Kano hadn't invented the judogi (Judo uniform) yet, and his students were still practicing in kimono. They were the wide obi still worn with formal kimono. In 1907, Kano introduced the modern judogi and its modern obi, but he still only used white and black belt ranks. The white uniform represented the values of purity, avoidance of ego, and simplicity. It gave no outward indication of social class so that all students began as equals. The black belt with the white gi represents the polarity of opposites, or In and Yo. The student begins empty, but fills up with knowledge.
Professor Kano was an educator and used a hierarchy in setting learning objectives for Judo students, just as students typically pass from one grade to another in the public school system. The Judo rank system represents a progression of learning with a syllabus and a corresponding grade indicating an individual's level of proficiency. Earning a black belt is like graduating from high school or college. It indicates you have achieved a basic level of proficiency, learned the fundamental skills and can perform them in a functional manner, and you are now ready to pursue Judo on a more serious and advanced level as a professional or a person seeking an advanced degree would. Of course, the rankings also represent progress towards the ultimate objective of judo which is to improve the self not just physically, but morally as well.
Around 1930 the Kodokan created a new belt to recognize the special achievements of high ranking black belts. Jigoro Kano chose to recognize sixth, seventh, and eighth degree black belts with a special obi made of alternating red and white panels (kohaku obi). The white color was chosen for purity, and red for the intense desire to train and the sacrifices made. The colors red and white are an enduring symbol of Japan, and they have been used in Judo since Jigoro Kano started the first Red and White Tournament in 1884. The kohaku obi is often worn for special occasions, but it is not required to be worn at any time and the black belt remains the standard for all the yudansha ranks. In 1943 the Kodokan created the optional red belt to recognize 9th and 10th degree yudansha.
Theoretically the Judo rank system is not limited to 10 degrees of black belt. The original english language copy (1955) of Illustrated Kodokan Judo, by Jigoro Kano, says: "There is no limit…on the grade one can receive. Therefore if one does reach a stage above 10th dan… there is no reason why he should not be promoted to 11th dan." However, since there has never been any promotion to a rank above 10th dan, the Kodokan Judo promotion system effectively has only 10 dans. There have only been 15 10th dans awarded by the Kodokan in the history of Judo.
Other colored belts for students who had not yet achieved black belt originated later, when Judo began being practiced outside of Japan. Mikonosuke Kawaishi is generally regarded as the first to introduce various colored belts in Europe in 1935 when he started to teach Judo in Paris. He felt that western students would show greater progress if they had a visible system of many colored belts recognizing achievement and providing regular incentives. This system included white, yellow, orange, green, blue, and purple belts before the traditional brown and black belts.
The Judo practice uniform and belt system eventually spread to many of the other modern martial arts, such as aikido and karate, which adapted them for their purpose. Karateka in Okinawa didn't use any sort of special uniform at all in the old days. The kyu/dan ranking system, and the modern karategi (modified judogi) were first adopted by Funakoshi in an effort to encourage karate's acceptance by the Japanese. He awarded the first shodan ranks given in karate to Tokuda, Otsuka, Akiba, Shimizu, Hirose, Gima, and Kasuya on April 10, 1924. The adoption of the kyu/dan system and the adoption of a standard uniform based on the judogi were 2 of the 4 conditions which the Dai-Nippon Butokukai required before recognizing karate as a "real" martial art. If you look at photographs of Okinawan karateka training in the early part of this century, you'll see that they were training in their everyday clothes.
Promotion requirements for each rank vary according to the sensei and the national organization that you are affiliated with. There is no worldwide standard for each rank, although it is generally accepted that a blackbelt has had many years of practice and can perform at least the nage-no-kata, the gokyo-no-waza and the newaza techniques. For further information about promotion requirements see your sensei.
Here's how to tie your belt, proper sizing for a judogi, and some information about where to get a judogi. Here is another article on Origins of the Karate Rank System and on Belt Colors and Ranking Tradition.