How does the Facebook Camera App know who I am the first time I go into it?
This was covered by MG Siegler over at Paris Lemon last week, the explanation of how it does it is here.

Basically, iOS provides it's own Keychain functionality and if you know the Keychain key, then you can save and read data between apps that you write.  In this case, the Facebook Camera app is reading data saved by the Facebook app.