A chôme (rhymes with Marisa Tomei) is a division used in Japanese cities. Each city, town, or ward is divided into several named districts, and each district is divided into several numbered chomes, which are then divided into blocks ("ban") and further subdivided into lots ("go").
The headquarters of NTT, for example, is located at 2-3-1 Otemachi, Chiyoda Ward, Tokyo. To find this address, you would first locate Chiyoda Ward on a map, and then locate Otemachi. The first number is the chome: in this case, 2-chome. The second number defines the block the building is on, and the third defines the specific building. Once you've found the block (in this case, 3), you just walk around the block until you find the right house or building number (in this case, 1), and you're done.
Within each chome, there are maps posted to show where the different
blocks are located: usually, the map can be found next to the chome's
koban (police box), or by major intersections around the periphery of
the chome. If you don't have a map, you can also find the block by
walking around and looking at the telephone poles, which are labeled
with their chome and block numbers.
This rather peculiar system is used because most Japanese streets don't have names or numbers of their own. Advertising for businesses in Japan will usually include a small map showing the location of the office or showroom relative to train stations, bus stops, and major landmarks. Sometimes, people have to draw maps in applications for their school or employer, just to show the location of their house.
Another byproduct of this system is that you can't hop into a taxi, give the driver an address, and expect him to be able to find it (at least without the help of an expensive GPS navigation system, which most taxis have nowadays). Instead, you're expected to give him directions: go to X station, make a left here, a right there. Giving a cab driver an address will probably cause the taximeter to run up as he consults his street atlas or his dispatcher to locate the chome and ban.
Historically, Japanese addresses had a single number after the district name, which was probably more aesthetically pleasing but made navigation even more hellish.