Narita International Airport (NRT), situated about 60 km (37 Miles) from central Tokyo, is the main airport in Tokyo that handles international air traffic. Take the JR Narita Express (NEX) to Tokyo station (approx 1 hour). (All seats are reserved); Or JR rapid trains (approx 1 hour and 30 mins) (No seats reserved); Or take a Limousine bus to Tokyo station (approx 1 hour and 40 minutes); Taxis are very very expensive as the airport is a distance from downtown so avoid!
Haneda Airport (HND), situated about 16km (10 mile) from central Tokyo, mostly handles domestic air traffic. You can take the Tokyo Monorail to Hamamatsucho Station, where you can take the Yamanote Line to downtown Tokyo (approx: 30 minutes). Or take a train on Keikyu Railways to Shinagawa Station, where you connect to the subway that takes you downtown. (approx: 30 minutes) Limousine Buses are also available to take downtown (approx 45min). Taxis are reasonably priced for downtown.
Official Narita International Airport
Official Haneda Airport
Official Airport Limousine Bus