[Tanick]

Read the post again, will try to answer. Central London is technically two cities: City of London (business district) and the city of Westminster (shopping, government, Buckingham Palace etc). If you refer to 'the City', it means the City of London so you are talking about the business district. The two are next to each other, Westminster being to the west (the 'West End' is Westminster) and the City of London to the east. The East End is east of the City of London (east of Tower Bridge really) and used to be the area with factories, docks, markets etc but now hopelessly fashionable in parts.

Mayfair is big on shopping, galleries etc.

Its such a big and diverse city that its hard to explain. If you are staying for a holiday then go for the West End if you can, but transport is good so its easy to get into the centre from outside. I've travelled extensively in the US but its hard to make a comparison. Parts are like DC, Mayfair and Chelsea are a bit like Georgetown, but other parts are much more New York, even Baltimore. Hard to sum up in a single post. It is a very safe city, friendly people and lots to do. Check out 'Time Out' for listings.