The Tokyo Subway (chikatetsu) is a difficult thing to node properly, because (a) it's big, (b) it's hard to comprehend even when you see a map of it, and (c) it's REALLY big. Also decentralized. Let's start by addressing that part...
Who's in charge here?!
Subways in Tokyo are operated by two authorities. Tokyo Metro, a private company, operates most of the lines. It is the direct descendant of the Teito Rapid Transit Authority (abbreviated Eidan), a joint venture of the Tokyo Metropolitan Government and the Ministry of Land, Infrastructure and Transport established in 1941. Eidan became Tokyo Metro in 2004 but remains a joint governmental venture.
The other lines are run directly by the Tokyo Metropolitan Government. These lines are prefixed with the name "Toei," which roughly means "administered by Tokyo." The gist of the story is that Toei lines were built where the local government wanted to have a transit link, but where Eidan refused to build for whatever reason.
The practical difference between Toei and Metro is slight; the systems are fairly well-integrated, although connecting between the two often requires going through a ticket gate. One noticable difference between the two is in their route maps: Toei lines are slightly thinner than Metro lines on Metro maps, and vice versa. Toei lines also tend to be more expensive and less pretty-looking than Metro lines.
So what are the lines?
In order of their construction:
- Ginza Line - Tokyo's first subway line opened in 1927 between Asakusa and Ueno. Over the next 12 years, it was extended past Ueno to Nihombashi, Ginza, Toranomon, Omotesando, and Shibuya. It is now 14.3 km long. Being the oldest line, it is also closest to the surface; you only have to go down one short flight of stairs to access it.
- Marunouchi Line - First opened in 1954, and reached its current 27.4 km length in 1962. From Ikebukuro, it goes through Korakuen, Ochanomizu, Otemachi, Tokyo Station, Ginza, Kasumigaseki, Yotsuya, several stops in Shinjuku, and then splits at Nakanosakaue to terminate in Ogikubo and Honancho.
- Toei Asakusa Line - Built between 1960 and 1968. It goes from Oshiage to Asakusa, Ningyocho, Takaracho, Shimbashi, Daimon, Mita, Gotanda, and Magome for a total length of 18.3 km. The train from Haneda Airport to Narita Airport uses the Asakusa line for part of its journey.
- Hibiya Line - First opened in 1961, completed in 1964. From Kitasenju, it passes through Ueno, Akihabara, Ginza, Hibiya, Kasumigaseki, Roppongi, and Ebisu, before turning into the Tokyu Tohin line near Meguro. Total length 20.3 km.
- Tozai Line - The longest Eidan line at 30.8 km, it opened between 1964 and 1969. From Funabashi, it travels to Gyotoku, Monzennakacho, Kayabacho, Nihombashi, Otemachi, Kudanshita, Iidabashi, Waseda, and Takadanobaba, and ends at Nakano where it merges with the JR Chuo line.
- Toei Mita Line - Opened in 1968, completed in 1976 and extended in 2000. Starting in Meguro, it goes to Mita, Hibiya, Otemachi, Suidobashi, Kasuga, Sugamo, Itabashi, Shimura, and Takashimadaira. 22.5 km.
- Chiyoda Line - Opened in 1970. It starts at Ayase and travels south to Nezu, Jinbocho, Otemachi, Hibiya, Kasumigaseki, Akasaka, and Omotesando, ending in Yoyogi and turning into the Odakyu Line. JR trains travel from Ayase east to Toride and beyond. Total length is 24 km.
- Yurakucho Line - Opened in 1974 and completed in 1988. It runs for 28.3 km from Kiba to Tsukishima, Ginza, Yurakucho, Nagatachou, Ichigaya, Iidabashi, and Ikebukuro, splitting at Kotake Mukaihara and ending in Nerima and Shinrin Koen.
- Hanzomon Line - Opened in 1978. It is 10.8 km long, starting at Shibuya and serving Omotesando, Nagatachou, Kudanshita, Jinbocho, Otemachi, Suitengumae, Sumiyoshi and Oshiage. The Shibuya end turns into the Tokyu Denentoshi Line; the Oshiage end turns into the Tobu Railway.
- Toei Shinjuku Line - Opened between 1978 and 1989. From Shinjuku, it passes through Ichigaya, Kudanshita, Jinbocho, Ogawamachi, Hamacho, Morishita, Sumiyoshi, Ojima, and Mizue, ending in Monoyawata, 23.5 km later.
- Namboku Line - Opened between 1991 and 2000, it is 20.8 km and goes from Akabane Iwabuchi to Oji, Todaimae, Iidabashi, Ichigaya, Yotsuya, Nagatachou, Tameike Sanno, and Meguro.
- Toei Oedo Line - This line is a monster, at 40.7 km. It forms a ring around the city remniscient of an underground Yamanote Line, starting and ending in Shinjuku and passing through Yoyogi, Roppongi, Azabu, Tsukishima, Morishita, Kuramae, Korakuen, Iidabashi, and Ushigome, tracing an almost perfect oval around the Imperial Palace. It also breaks off into a spur from Shinjuku to Nakai and Hikarigaoka. Being the newest line in the network, it is also the deepest underground; one of my friends describes it as being "in the seventh level of Hell," which practically means that you might have to go down about five escalators to get to it, depending on how many underground lines already existed around the station in question.
- Fukutoshin Line - The newest line in the system opened in 2008 and connects Ikebukuro to Shibuya, basically running parallel to and just inside the Yamanote Line. The main point of this line is to bring through traffic from the Seibu Railway's Ikebukuro-bound lines to the Tokyu Electric Railway's Shibuya-bound lines, and to relieve overcrowding on the west side of the Yamanote. It is 20.2 km, about 9 km of which is shared with the Yurakucho Line.
Whoa! That's 283 kilometers of underground track! Can you believe it?
What about riding it?
Mass transit in Tokyo, I should emphasize, is NOT as bad as the stories you've probably heard. Ten or twenty years ago, the stories of super-overcrowding were apparently true. Now, they aren't: the network has matched capacity demands, or at least become closer than it was. There are pushers at the largest stations during the height of rush hour, but you will never encounter one if you sleep past 9 AM every morning. (I leave for the office around 9:45 now, and can usually get a seat on the train through the center city.)
Fares range from ¥150 to ¥300 each way, depending on how far you go on the network. The Toei and Eidan subways offer both separate and combined day passes: the Eidan even offers an unlimited monthly pass (for ¥17,000—do you really need it?)
Most lines operate from 5 AM to just after midnight. Avoid rush hours on weekdays, if you can.
Also, expect stations to be old and confusing. Japan is not renowned for its urban planning.
Tidbit: Tokyo Metro trains have little "mind the door" stickers which feature a little cat getting its tail crushed in the doors. When I see these, I can't help but think: "Every time you lean on the door, God kills a kitten."