Is it possible to write an android app using C#? Is there an API or something? Is it free for personal and commercial development?
What do I need to install to have
You could use Mono for Android:
Mono for Android
http://xamarin.com/monoforandroid
An alternative is dot42:
dot42
http://www.dot42.com/
dot42 provides a free community licence as well as a professional licence for $399.