Take the train. This is the quickest option in most cases, although it can be more expensive than taking a coach. Heathrow has 2 train services that run into London Paddington, the Heathrow Express and the slightly slower Heathrow Connect. The Gatwick Express runs into Victoria station; Southern Trains and First Capital Connect also run train services from Gatwick. These services have stops but are cheaper. Stansted also has the Stansted Express Line. Luton is the only airport not to have a dedicated train station; instead, you must take a shuttle bus to the Luton Airport Parkway station. From there, you can choose from First Connect and East Midlands trains.
Buy a ticket for a coach. Coaches connect all 4 airports to London and are generally cheaper (if slower) than trains. The National Express coach service travels into London from all airports. Easybus services Gatwick, Luton and Stansted.
If you don't mind sharing, make a reservation with National Express's Dot2Dot, which takes arrivals into London in vans that hold 10 to 12 people. Dot2Dot runs from Heathrow into central London and Canary Wharf, and it is significantly cheaper than a taxi.
If you're flying into Heathrow, take the Underground into London for a low fare. There are 3 stations on the Piccadilly line: one for Terminals 1, 2 and 3, one for Terminal 4, and another for the newest terminal, Terminal 5. The trip takes about an hour. Since you will be boarding at the end of the line, there should be room for your luggage, although you may have to navigate through a crowded train by the end of your journey.
Ride the bus. If you're arriving late into Heathrow, you find the Underground closed, and you don't have cash to spare, the Number 9 bus route runs into London day and night. The trip takes a while, but the bus route does end at Aldwych, in the heart of central London.