Berlin has three airports, two for international flights, and one mainly for domestic flights. They are supposed to be merged at some time in the future into one large airport (they say 2011. You can read all about it here).
Until then, you have Schoenfeld (SXF) in the east (some 20km from the center), Tegel (TXL) in the middle (some 8km from the center) and Tempelhof (THF), which is supposed to be the oldest commercial airport in the world (built in 1923) and the first airport connected to a subway network (1927), but is not profitable, and scheduled to close on 2008. Anyway, it does only domestic flights within Germany and neighboring countries.
In our first visit we landed in Schoenfeld. We got out quickly, and the public transportation to the city was accessible. The ride was a bit long. Leaving was another issue, but it had to do with the fact that we traveled with an Israeli airline, Israir (a very bad one, in case you ever consider to travel with them), with all their paranoid security checks and lockdowns and other inconveniences.
The second time we traveled with TUI‘s Hapag Fly (Hapag Lloyd), which was a much more pleasant experience, and we landed in Tegel. Very accessible airport, very well designed, on the whole – very comfortable.
Schoenfeld to Berlin
To get from Schoenfeld to Berlin you take the Airport Express – it stops at all the main stations, such as Ostbahnhof, Alexanderplatz, Hauptbahnhof and Zoologischer Garten.
The S9 train also gets you to most of the central stations. The S45 goes to more esoteric places. You also have bus lines 160, 163, 171, 602, 735, 736, 737, 738, N46, N60, but trains are a better means of transportation here. Taxi ride, they say at the official city site, will cost you around 23 euros.
Tegel to Berlin
From Tegel you have the TXL bus, which is quite convenient, and also the bus lines 109, 128, X9.
You can get public transportation maps at the airports. You may also want to buy a WelcomeCard, daily ticket or weekly ticket for the public transportation at the machines there.
For more information, check out the Berlin airports site.