Selfridges on Oxford Street is great, it has things at a range of prices as someone else said
And also Harrods just because most people who come to London want to go there. That's in Knightsbridge
Covent Garden is also lovely, quite a few places to eat and unusual shops. Also a few known ones, I'm sure there's an Urban Outfitters and Mac nearby, although I havent been to C.G. for ages...
Also Browns which is on (I think?) South Molton St off Oxford St. It's like a boutique-y place, they have a website. Then Bond St. has lots of designer boutiques, Hermes, etc
Very nearby is Regent Street which has yet more shops
All these streets are quite close together so you can basically walk from one to the other, just go to google maps and print off a map and you'll know what I'm talking about. I hope this helps, have a great time!
