libc implements both standard C functions like strcpy() and POSIX functions (which may be system calls) like getpid(). Note that not all standard C functions are in libc – most math functions are in libm.
You cannot directly make system calls in the same way that you call normal functions because calls to the kernel aren’t normal function calls, so they can’t be resolved by the linker. Instead, architecture-specific assembly language thunks are used to call into the kernel – you can of course write these directly in your own program too, but you don’t need to because libc provides them for you.
Note that in Linux it is the combination of the kernel and libc that provides the POSIX API. libc adds a decent amount of value – not every POSIX function is necessarily a system call, and for the ones that are, the kernel behaviour isn’t always POSIX conforming.
libc is a single library file (both .so and .a versions are available) and in most cases resides in /usr/lib. However, the glibc (GNU libc) project provides more than just libc – it also provides the libm mentioned earlier, and other core libraries like libpthread. So libc is just one of the libraries provided by glibc – and there are other alternate implementations of libc other than glibc.