Both Windows (Win32 API) and OS X (Cocoa) have their own APIs to handle windows, events and other OS stuff. I have never really got a clear answer as to what Linux’s equival
To aid in what has already been mentioned there is a very good overview of the Linux graphics stack at this blog: http://blog.mecheye.net/2012/06/the-linux-graphics-stack/
This explains X11/Wayland etc and how it all fits together. In addition to what has already been mentioned I think it's worth adding a bit about the following API's you can use for graphics in Linux:
Mesa - "Mesa is many things, but one of the major things it provides that it is most famous for is its OpenGL implementation. It is an open-source implementation of the OpenGL API."
Cairo - "cairo is a drawing library used either by applications like Firefox directly, or through libraries like GTK+, to draw vector shapes."
DRM (Direct Rendering Manager) - I understand this the least but its basically the kernel drivers that let you write graphics directly to framebuffer without going through X