London has 10 airports if you include the smaller domestic ones (one is actually a helipad) but if you are looking at international airports there are six.
By far the biggest and most important of these is Heathrow, which has five terminals and two runways. In fact, for international passenger numbers Heathrow is the busiest airport in the world, and for overall numbers it is still the third busiest. It is also one of the world's most overloaded and crowded airports; there have been many plans to solve this, but it is difficult to expand Heathrow due to its position.
The second largest London airport is Gatwick. Gatwick has two terminals and is one of the busiest airports in Europe.
The remaining London-based airports are Stansted, Luton, London City and Southend. Southend is smaller than the others but has quite a lot of low cost flights.

You can learn more about the major London airports here.