Asakusa - Old Tokyo, where the Kaminarimon and the large temple are
Odaiba - The artificial island in Tokyo harbour. Go spend some time at the Oedo Onsen Monogatari, which is the traditional Japanese bath-house is. Also, Fuji-TV is there. It's an awesome building.
Akihabara Electric Town- The electronics/arcade district. Also home of Yodobashi, the largest electronics store in Tokyo. There's also a good conveyor belt sushi place in the main JR station

Harajuku - The youth fashion district. Brand name stores and teens dressed in insane outfits heh
Ginza and Shinjuku - the more mature modern shopping districts. Mitsukoshi is in Ginza, where the world's first department store was set up. Oh and if you're in Shinjuki, careful not to wander into the red light district, unless of course you're into that stuff lol.
Ueno - Ueno Park is the thing I remember about this area. Plus it was where I stayed. Cheap Ryokan for 2500 yen a night!
Ikkebukuro - The cheap clothes and shopping district. Engrish clothing EVERYWHERE. Plus my favourite anime paraphrenelia store, Anime, was located there, if you're into that stuff hahah.
Roppongi - Bar district. And the Japanese girls there love the white folk heh.
I could seriously go on forever. Tsukiji fish market, the parks. Tons to do. How long do you have there?