When Londoners talk about 'the City', or road signs point to 'City', they invariably mean the City of London, which is the area occupied by the ancient walled City. This is now London's principal business and financial district. The main shopping areas, and most of the theatres, are in the area to the West of the City, always described by Londoners as the 'West End'.
The City covers an area of just one square mile - and is often referred to as 'the Square Mile' in conversation. Apart from the financial institutions, St Paul's Cathedral, the Museum of London, and the Barbican arts centre all fall within this area, and the Tower of London is right on the edge of it. A few thousand people also live in the City, mostly in the flats in the Barbican. The City has its own system of local government and, strangely, its own police force. The local authority for the City is officially called the 'Corporation of London', though it has no jurisdiction in the rest of London.
To confuse matters, most of the West End falls within an area known as the City of Westminster, an area which extends North and West past Paddington Station. In everyday usage, 'Westminster' is used to describe just a small area around the Houses of Parliament.
'Central London' is probably the best generic term for the City and the West End together. But it is rather an ill-defined concept, maybe reflecting the extent to which London has grown up as a series of connected villages. As you approach London road signs normally point either to 'City' or to 'West End'.
'Greater London' refers to the whole of London, including most of the suburbs, and was a term introduced in 1965. From then until 1986 this was the area covered by the Greater London Council. Apart from the Cities of London and Westminster, it is divided into a number of 'boroughs'. Greater London is usually shown as a county on maps of Britain.
The area covered by the Travelcard scheme is Greater London. A few stations, such as Hampton Court, and Elstree, which are outside Greater London are actually in the Travelcard area because the districts that they serve are part of Greater London.
Some place names are not pronounced the way that you would expect. Greenwich, Holborn, Clerkenwell, and Leicester Square are pronounced Grennitch, Hoe-bun, Clarkenwell, and Lester Square respectively.
In contrast to much of the rest of the English-speaking world, street names in London - or anywhere in Britain - are rarely abbreviated by dropping the 'road', or 'street' from the name. On the contrary, major streets are often referred to with an extra 'the' - as in the Fulham Road, or the Tottenham Court Road. A few street names, such as Knightsbridge and Piccadilly are complete in themselves and never take a 'street' or 'road'.
This is particularly important in referring to tube stations and bus stops, and in stating your destination to taxi drivers. The tube stations of Gloucester Road, Warwick Avenue, and Baker Street would never be referred to just as Gloucester, Warwick, or Baker. The story of the tourist who wanted to go to Liverpool Street station in London, but ended up on a train to the city of Liverpool, 200 miles away, is probably an urban myth. But it illustrates the point. More likely to cause confusion in practice, there are two Edgware Road stations, on different lines, but within a few hundred yards of each other. Both of these are a long way from the station called Edgware.
If you address a letter to somebody and purely put 'London' after the street address, it is unlikely to reach its destination. You will need to at least specify the postal district, which appears as the first part of the postcode. For example my office has a postcode of EC2Y 8HB; the postal district is EC2 (EC stands for East Central). The postal district where I lived for many years is SW15 (SW for South West). Postal districts have been around for a lot longer than postcodes; apparently they were introduced during the first world war as a way of coping with the volume of mail sent home by the troops. The division of London into various segments (such as SW, EC, and so on) has been used for longer still, so documents from the nineteenth century may refer to addresses in, for example, 'London SW'. When postcodes were introduced in the 1960s, those for London were generated simply by adding some extra information to the existing postal districts. Street signs, and the indices in street atlases of London, usually have the postal district written after the name of a road.
Within the segments, the numbers are not allocated geographically, except that SW1, N1, and so on are all close to the centre of London. Most of the other numbers are allocated alphabetically within certain ranges. So for example in South West London there is a block of postal districts running from SW11 (Battersea) to SW20 (Wimbledon West).
The London postal area is much smaller than Greater London. This means that there are many places which are administratively in London, but whose postal addresses are in one of the adjoining counties. For instance Richmond is part of London, but letters there must be addressed to Richmond, Surrey. This has caused a particularly strange anomaly. The county of Middlesex, just North of London, ceased to exist when Greater London was created in 1965. But it survives as a postal address, covering an area North and West of London which the Post Office cannot easily incorporate into any other county.