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London Heathrow Airport
Heathrow is one of the largest and busiest airports in the world. It is probably the best place in Europe to get a scheduled flight anywhere in the world (particularly North America) since the sheer number of flights to different destinations means that cheap deals are far more often available from here than from other places. There are no budget airlines flying from here however, and cheap deals from airlines such as BMI (as opposed to their low-cost subsidiary BMI Baby) are often slightly negated by high airport charges. It is fairly easy to get here from other London airports or stations however.
The airport itself consists of four (soon to be five) large terminals. Terminals 1-3 are all connected and served by the Heathrow Central bus, tube and railway stations. Terminal 4, and 5 when it opens, are some distance away and are connected by Heathrow Express. All the terminals are large, which means that there are generally plenty of facilities of all kinds there, though they are also busy for most of the day. Check in early as there can be queues at security, and the walking distance to many of the gates is pretty significant.
Most of all this is a really busy airport. If crowds and chaos make you uncomfortable and flying from Gatwick is an option, you might want to do that instead.
Sleeping there
Heathrow airport is open 24 hours, although all the shops and facilities tend to close for a few hours at night. In principle it should be no problem to sleep there if you have a ticket for the next day, but at such a key location you may well find bored security staff and armed police will hassle you constantly.
Getting there on foot
Heathrow airport, and associated hotels, airline offices and trading park take up a vast area of land at the western edge of London. This is hemmed in by the M25 and M4 motorways. It is perfectly possible to walk there from Feltham or even Richmond, and probably also from Hayes (central London is too far). It is probably a little disheartening though since there are cheap bus services from all of these places which will drive past you every few minutes day and night. They will probably drive through a puddle and splash you. You don't want to arrive at check-in late and soaking wet do you? You'll look ridiculous.
Getting there by public transport
Tube.
The easiest and cheapest way from central London is to get the tube. Heathrow is at the western end of the Picadilly Line (30 minutes, every 5-15 minutes). The fare is 4 pounds, but can be as little as two pounds if you get an Oyster Pre-pay card from the tube station before you travel. A zones 1-6 travelcard (6.30 pounds off-peak) will cover this journey and also allow unlimited tube, bus and train travel in Greater London for the rest of the day. Currently the tube only runs to terminals 1-3, but there is a free shuttle bus running from Hatton Cross station to terminal 4 (listen for announcements on the train). Full details of this service are on the Transport For London website.Bus.
If you arrive between midnight and 5am the tube will not be running, and you should get the N9 night bus (40-50 minutes, every 20 minutes). If you want to get to south or west London, you may be better off catching a bus instead of the tube. In particular the airport express service runs regularly to Feltham and Richmond. Local buses cost 1.20 per journey (0.80 if you get an Oyster card), and are free if you have a daypass (3.00) or any kind of travelcard. All buses go from the Heathrow central bus station, which is in the central area between terminals 1,2 and 3. Again full details are on the Transport For London website.Train.
The Heathrow Express train service takes 15 minutes to reach central London. It is much more expensive than the tube however, and arrives at Paddington which is the least central of the London stations. Basically this is probably not generally worth considering. It is free however to use these trains between terminal 4 and terminals 1-3.The Railair coach runs from Heathrow central bus station to Reading Station (45 minutes, every 20-30 minutes). The basic fare to Reading is 12 pounds, but integrated tickets are available to stations on the National Rail network. Buy tickets before you board from the ticket counters at the central bus station (or the usual National Rail ticket outlets).