A file descriptor is an opaque handle that is used in the interface between user and kernel space to identify file/socket resources. Therefore, when you use open() or socket() (system calls to interface to the kernel), you are given a file descriptor, which is an integer (it is actually an index into the processes u structure - but that is not important). Therefore, if you want to interface directly with the kernel, using system calls to read(), write(), close() etc. the handle you use is a file descriptor.
There is a layer of abstraction overlaid on the system calls, which is the stdio interface. This provides more functionality/features than the basic system calls do. For this interface, the opaque handle you get is a FILE*, which is returned by the fopen() call. There are many many functions that use the stdio interface fprintf(), fscanf(), fclose(), which are there to make your life easier. In C, stdin, stdout, and stderr are FILE*, which in UNIX respectively map to file descriptors 0, 1 and 2.