The following passage isn't meant to be any sort of complete description of how libraries, compilation processes or system calls invoking works but rather a bird-eye view of what OP asked, thus lacks several details and important passages which will have to be studied in-depth by the OP himself
By "C++ library" I assume you're referring to the C++ standard library (although the considerations here are valid for any other library as well).
The C++ standard library isn't mandatorily present on any operating system by default, it usually comes shipped with a compiler installation or with a secondary package. That doesn't mean you can't execute C++ compiled routines, you just need headers and the library in order to compile your programs along with a compiler which supports it.
The C++ standard library is usually compiled platform-specific and you can't just copy the headers and lib files from one operating system to another one (you'll end up in tears).
Everytime you import the declarations from a header file with something like
#include <iostream>
you're rendering your program aware of the moltitude of data structures, functions and classes provided by the standard library. You can use them as you want as long as you provide the .lib file (in a Windows environment) where the code of the routines is usually defined (in Visual Studio this is usually referred as the Runtime Library with /MT /MD
options) for linking.
Once you've linked your executable against those .lib files, you have a compiled executable which, opened in a disassembler, might have something like (for a simple hello world, snippet from here - not a windows environment)
mov edx,len ;message length
mov ecx,msg ;message to write
mov ebx,1 ;file descriptor (stdout)
mov eax,4 ;system call number (sys_write)
int 0x80 ;call kernel
thus eventually every C++ function or routine provided by the standard library either implements an algorithm and/or eventually call some operating-system specific routines through System Calls. There are several design and implementation differences between the various operating systems (and even the boundaries of the system call points) in addition to a thousand of layers for security checking (not to mention ring3/ring0 switches) so I won't spend more words here about that.