You can buy almost anything you can think of in the shopper's paradise that is Tokyo, but one of the most popular things that people come home with, aside from confusion is a big pile of electronic goods that are far cheaper than they would have been in the UK, and Shinjuku is the place to go in order to pick them up. Beware that Japan uses 100v AC for power and unless the items are certified as being dual voltage, or you buy a rather unwieldy adaptor, they're as good as useless when you get back to Britain with them. Also make sure that you get a manual in English, and an international guarantee from the vendor.
Ginza is the area of the city where you can find the glitziest department stores, and Shibuya is great for clothing. If you want to pick up some Japanese style handicrafts such as Kimonos, artwork, or even sumo nappies, then head to the small stores around Asakusa Temple where there are a few markets specialising in them.
Ameyoko Market is worthy of mention as it was the centre of the Tokyo black economy during the Second World War, and you can still pick up plenty of things there cheaper than in the rest of the city.
Shops are open from around 10am until 8pm seven days a week, and if you ask politely, you may get a discount, although courtesy is so important during negotiation in Japan that you need to say thank you afterwards or they will put the price back up.