Graphics API
A graphics API is how a game gets its graphics card to do its work, it renders the image you see on the screen. Some API's can make a game look nicer, but they are more hardware demanding for you device so they will need more powerful hardware around them to work properly, such as DX11 for example, it will show the game as really pretty, but if your computer is not powerful enough to use it (by computer I mean the hardware that makes the computer altogether just do things, like the CPU and how much RAM you have etc.), the frames per second is going to be bad/ low. Whereas other graphics APIs will not look as good, DX9 for example. But with not looking as good, they will be easier to run and therefore will have a higher FPS, that is if the FPS is not optimal when you use a graphical API such as DX11. (When I say DX I mean DirectX, the different numbers are different versions of it, as you can imagine, DirectX 9 is older than DirectX 11).
OpenGL
Usable with many platforms and operating systems, released in 1992, it was created as an open and all round baseline/ standard thing for everyone and everything. Device independent as all hell.
Microsoft DirectX
Built into Microsoft windows, I think it's safe to say that it is not quite as platform independent as OpenGL, DirectX is special because it is not just a graphics processor but like... a multimedia processor, so it is able to work with a lot of different kinds of things, but only on windows.