Implementation is the actual code with the logic that does the thing you want it do, as opposed to the command, which is how you tell the system what it should do.

The command can be the same on multiple OSs, but the implementation can be different.

In case of Linux and the coreutils (which are the basic programs you need beside the kernel to make a functioning system, stuff like mkdir) the most common implementation of all the coreutils is the one made by GNU. Stallmann did a lot of work on that so he wants credit for making a big part of the OS.