If it is for sale anywhere in the world, you can probably buy it in Tokyo. Tokyo is one of the fashion and cosmetic centers in the Eastern world. Items to look for include electronics, funky fashions, antique furniture, and kimono, as well as specialty items like Hello Kitty and Pokemon goods, anime and comics and their associated paraphernalia. Tokyo has some of the largest electronic industries in the world, such as Sony, Panasonic, and Toshiba, etc.
Cash payment is the norm. Most Japanese ATMs do not accept foreign cards, but the post office, 7-Eleven, and Citibank ones do and usually have English menus as well
Although credit cards are becoming more widely accepted, retailers are much less likely to allow their use than in most other developed countries. The crime rate is very low, so don't be afraid of carrying around wads of cash as the Japanese do. The average Japanese citizen will carry a month's worth of expenses on them.
There are numerous convenience stores throughout Tokyo, which are open around the clock and sell not only food and magazines but also daily necessities such as underwear and toiletries. Supermarkets are usually open until 22:00, while drugstores and department stores usually close at 21:00.
Anime and manga
Akihabara, Tokyo's Electric Town, is now also the unquestioned center of its otaku community, and the stores along Chuo-dori are packed to the rafters with anime (animation) and manga (comics). Another popular district for all things manga/anime is the Nakano ward and its Broadway Shopping arcade. Check out the mandarake shop for loads of used and rare mangas.
In recent years there has been an "otaku boom" in Akihabara. A lot of attention, in particular, was paid to the town thanks to the popular Japanese drama "Densha Otoko", a love story about an otaku who saves a woman on a train and their subsequent courtship.
Akihabara was previously known for its many live performances and cosplayers, some of which had drawn negative attention due to extremist performers. These have become increasingly scarce following the Akihabara massacre in 2008, although girls in various maid costumes can still be seen standing along the streets handing out advertisement fliers to passers-by for Maid Cafes.
Antiques
Serious collectors should head for the Antique Mall in Ginza or the Antique Market in Omotesando, which despite the rustic names are collections of small very specialist shops (samurai armor, ukiyo-e prints, etc) with head-spinning prices. Mere mortals can venture over to Nishi-Ogikubo, where you can pick up scrolls of calligraphy.
The Antique Festival (全国古民具骨董祭り) is held over the weekend about 5-6 times a year at the Tokyo Ryutsu Center, on the Tokyo Monorail line, and is well worth a visit.
Books
Jinbocho is to used books what Akihabara is to electronics. It's clustered around the Jinbocho subway stop. The Blue Parrot is another shop located at Takadanobaba on the Yamanote line, just two stops north of Shinjuku.
Cameras and electronics
Ever since Sony and Nikon became synonymous with high-tech quality, Tokyo has been a favored place for buying electronics and cameras. Though the lines have blurred since the PC revolution, each has its traditional territory and stores: Akihabara has the electronics stores, including a large number of duty-free shops specializing in export models, and Shinjuku has the camera stores. Unfortunately, local model electronics are not cheap, but the pre-tax prices for the export models are similar to what you'll pay pre-tax in Europe and are a little higher than U.S. prices. You can sometimes find cheap local models if you avoid big shops and check smaller retailers, and are willing to deal with Japanese-only interface, manual and service warranty. It's also surprisingly difficult to find certain things, eg games machines.
Fashion
Shibuya and neighboring Harajuku are the best-known shopping areas for funky, youthful clothes and accessories. Note that, almost without exception, clothes are sized for the petite Japanese frame.
Department stores and exclusive boutiques stock every fashion label imaginable, but for global labels prices in Tokyo are typically higher than anywhere else in the world. The famous Ginza and Ikebukuro's giant Seibu and Tobu department stores (the largest in the world) are good hunting grounds. Recently, Roppongi Hills has emerged as a popular area for high-end shopping, with many major global brands. Other department stores in Tokyo are Mitsukoshi, Sogo, Marui (OIOI), and Takashimaya. Mitsukoshi is Japan's biggest department store chain. Its anchor store is in Nihonbashi. Marui Men store in Shinjuku has eight floors of high-end fashion for men only.
Kitchenware
The district for this is Kappabashi Street near Asakusa, also known as “Kitchen Town.” The street is lined with stores selling all kinds of kitchen wares — this is where the restaurants of Tokyo get their supplies. It's also a great place to find cheap Japanese ceramics, not to mention plastic food!
Music
Ochanomizu is to the guitar what Jinbocho is to used books. There, you’ll find what must be the world’s densest collection of guitar shops. Plenty of other musical instruments (though not traditional Japanese ones) are also available.
Adult goods
Japan is a world leader in various intimate adult products, from stylish Tenga devices to various dolls with the replaceable body parts (and whole floors selling the spare "parts" for them), as well as some other bizarre goods. Akihabara hosts several large, multi-floor adult stores which are clearly marked as such. Prices for the Japan-made goods are generally better than in Europe and the US; notably, Tenga products are 10-12% cheaper than the lowest online price in the US or Europe, and the selection is better. If buying toys or lingerie make sure you purchase the "export" version to avoid size issues.
Souvenirs
For touristy Japanese knickknacks, the best places to shop are Nakamise in Asakusa and the Oriental Bazaar in Omotesando, which stock all the kitschy things like kanji-emblazoned T-shirts, foreigner-sized kimonos, ninja outfits for kids and ersatz samurai swords that can be surprisingly difficult to find elsewhere. Both also have a selection of serious antiques for the connoisseur.
Street markets
Bustling open-air bazaars in the Asian style are rare in Tokyo, except for Ueno's Ameyoko, a legacy of the postwar occupation. Yanaka Ginza in the Shitamachi Taito district, a very nice example of a neighborhood shopping street, makes for an interesting afternoon browse.
There are often small flea and antique markets in operation on the weekend at major (and minor) shrines in and around Tokyo.