Strictly speaking, the API of Linux consists of its system calls. These are all of the kernel functions that can be called by a user-mode (non-kernel) program. This is a very low-level interface that allows programs to do things like open and read files. See http://en.wikipedia.org/wiki/System_call for a general introduction.
A real Linux system will also have an entire "stack" of other software running on it, in order to provide a graphical user interface and other features. Each element of this stack will offer its own API.