I own an old 2004 Apple Cinema Display 20" (Aluminium) M9177LL/A and just bought a 2012 Macbook Air. When I'm at home, I'd like to connect the Cinema Display to my closed Macbook Air, but don't know what adapter to buy. The old Apple Cinema Display has an unusual connector.

Any idea what I need to buy?

Your aluminum 20-inch Cinema Display should have a standard DVI connector.
What makes you think that's unusual?

And, you should be able to connect that to your MacBook Air with this: http://store.apple.com/us/product/MB570Z/A
which is a Mini DisplayPort to DVI Adapter