Tokyo has several districts (or neighborhoods) that are known for specific types of shops or activities. Most are known by the name of the nearest train station. The map here is not to scale, but it's based on the standard JR train line map. The Yamanote loop is about 5 kilometers wide, and 10 km long. The Chuu-ou line is about 4 km from the north end of the loop and runs the full width.
Shinjuku: A major nightlife area. The west side has electronics stores and game arcades. The station itself has several major department stores, as does the south exit. The north side leads to more stores, restaurants and movie theaters. The east side has Shinjuku Gyoen, and lots more stores, shops, and bars. The north exit street area is popular in manga, with the big TV screen on the Studio Alta building. (If you wanted to, you could go just a short ways north from the station to Yasukuni street, and then just follow it east to Jimbocho (see below).
Yoyogi: Has Yoyogi park, and the imperial royal museum. Near the park is an NHK TV studio.
Harajuku: On the Yamanote line, between Shinjuku and Shibuya. On the east side are the fashion shops aimed at teenage girls. On the west is a continuation of Yoyogi park, and the main drag that is popular with indie bands on Sundays.
Shibuya: A major nightlife area, with multiple department stores, movie theaters and fashion shops. A primary hangout for teenage girls. Right in front of the station is Hachiko, a statue of a dog that used to visit the station a long time ago - Hachiko is an easily-recognizable landmark for people waiting to meet up with their friends. The intersection here is the one that always shows up in anime - with the diagonal crosswalk and big TV screens. One of the department stores here is Parco, which has an art gallery on floor 6 of building 1. This is where the Tezuka Gene exhibit was held. In the basement is Logos, the bookstore that includes a small show space that had the Hell Girl display.
Ebisu: Beer garden central. Home of the Yebisu brewery's Yebisu Beer Museum.
Jimbocho: The place to go for used books. It's right on the outskirts of Kanda, 1-2 kilometers west of the JR Kanda station on the Yamanote line, and therefore people often refer to the area as being the "Kanda used book district", although this is not really correct. The majority of the bookstores are on Yasukuni Dori (Yasukuni Street), running on the east side of the Toei Shinjuku subway line's Jimbocho station.
Ochanomizu: Just south of the Ochanomizu station on the Chuu-ou and Sobu lines, there is "the musical instrument area". This is a series of shops that specialize in one kind of instrument, along Meidai Dori. About 10 shops alone carry electric guitars. That's lots of guitars. Across the street is a massive place carrying only violins. On the same street is Meiji University.
If you go down Meidai Dori from Ochanomizu, past the music instruments shops, you'll hit Yasukuni Dori. Go west and you get to Jimbocho station (keep going west and you'll hit Shinjuku). Go east and you enter the sports shops district. This is a series of large stores that carry sporting goods, such as outdoors gear, bicycles and the like. In December, over ten shops had nothing but snowboards set out on the sidewalk for sale. This district is over 4 blocks long and is packed with pretty much nothing but sporting goods shops.
Ueno: Near the JR Yamanote line Ueno station are lots of little shops with import foods. Plus, Ueno Zoo.
Roppongi: This is probably one of the most famous areas in Tokyo, not just because of all of the night clubs and drinking spots, but also because it is the home for most of the foreign embassies (a number of other embassies are scattered around the Yamanote loop). The U.S. embassy is here. Roppongi is just east of Ebisu and south of Shinanomachi, bordering on Tokyo Midtown. No idea why diplomats would want to be so close to that many bars and young people...
Yurakucho: A major nightlife district with food stalls, bars and movie theaters, bordering right on Ginza. Reachable by taking the Yamanote line 1 stop southwest of Tokyo station.
Ginza: Very upscale neighborhood. You name it, it's here (not including parks, though). Mikimoto, the famed seller of pearls, has its display store here. Just take the Yamanote to Yurakucho and head east 2-3 blocks.
There are other specialty districts in Tokyo, but I'm running out of space to list them. Maybe later.