Fantastic info from londonkiwi, I second it all!
Though Oxford Street and Regent Street are indeed the "main shopping drags" here in London, you don't really need to go near either for a good day of shopping. Unless I need to visit one of the main department stores, I try to avoid the West End at all costs- it's so crowded!
I like shopping near Covent Garden/Long Acre, which lk also mentioned. It has most of the big chain shops. For a change of pace, I like to go to High Street Ken ("Kensington High Street" tube) for my shopping- loads of chain and indie shops and even a Topshop and M&S, too. The King's Road in Chelsea is great, too. All the higher-end high street shops are there, and there's a big John Lewis (confusingly called Peter Jones, but the same thing) at one end.
If you're looking for a nice day of sightseeing, good food and a bit of shopping, you can't do better than Hampstead Village in north London. The heath is great for walking, there are some art galleries and places of interest, and loads of great cafes and restaurants. The shops there are a mix of high street and lovely independent boutiques- I've found great stuff here. Marylebone High Street has a similar feeling (but without the fabulous heath).
In terms of shops, lk mentioned a lot of the ones we have- H&M, Topshop, Zara, Gap. When friends visit me, though, I try to take them to places that are only in the UK, or at least much easier to find here. With that in mind, some of my favourite places to shop are:
Gap- sounds strange to start with this, but the merchandise is different in the UK. There is a London-based design team for the Euro stores, and about 90% of the merchandise in unique to the stuff in North America. It tends to be a bit more fashion-forward, so it's always worth checking out.
Whistles- my absolute favourite shop! Feminine, often retro, great use of print and colour. Their new design director, Jane Shepardson (of Topshop fame) has really stepped up the fashion factor and the new season's merchandise is FAB.
Jigsaw- I also love Jigsaw, which has a slightly bohemian, arty, intellectual feeling to its collections. The brand uses colour really well and I always like the fall collections. Jean, I've always thought of you as a Jigsaw girl, because of your colouring and artistic sense Has a less expensive and "softer" little sister brand called Kew which is nice, too (there's one at High Street Ken).
Reiss- gorgeous clothing. Very upper end of the high street in terms of prices, but excellent quality too. Very much led by high fashion.
COS- H&M's high-end, more mature line, though you'd never guess because it's branded completely independently. Very edgy, with the focus on interesting shapes in basic solids (not a lot of print). Even better for men than women- especially slim ones. There are two shops in London, one on Regent Street and one on Long Acre in Covent Garden.
LK Bennet- makes great shoes but nice clothing, too (one on the King's Road)
Mexx- I haven't shopped here for years but I know Angie is a fan, so it's got to be good!
Hobbs- From the outside, this place looks a bit stale and "old lady", but they make nice clothing. I've never bought anything, but my sister gets nice suits and wrap dresses here, and they always have great retro-style dresses in the window that catch my eye.
Massimo Dutti- I think this brand is Italian, but I'm not sure where else they are in the world. I've shopped there occasionally- some nice stuff.
Uniqlo- obviously not British but I know they're not everywhere in the States, and I have good luck here. Great for skinnies and V-neck cashmere sweaters in every colour of the rainbow.
Agnes B- Lovely modern but nostalgic French brand, worldwide but obviously not in every city. There are branches in Hampstead and Marylebone Village.
Hmmm, I think that's all I can think of for now. I'm sure more will come to me, though! If you're interested in food, I'll second lk on the Borough Market rec! This is one of my favourite places in London, and though I shop more at my local farmer's market, I love going here every now and then. The best coffee shop in the city, Monmouth Coffee, is right next to the market (there is another in Covent Garden on Monmouth Street). There is usually a line but it's worth the wait!
