Hey there!
From what I understand, a 'standard' or single-user Operating System operates one computer, for example, I have an Apple computer, and my OS is Mac OSX (A lot more people on here probably use Windows as their OS)
This OS is on a single computer, and controls only that computer, although it still allows network connections to other computers, even ones with different OS's installed.
A network OS, I think Linux is very commonly used, is installed on big, huge super-computers (Servers, as they are known) that many people on a huge network (A call centre in SKY TV or BT) access. Normally, each person at the desk normally has a Screen, Keyboard, and Mouse, all wired to one Server, that has one OS installed, that everyone can access at once, via their own screen.
I think thats correct, but please correct me if I am wrong!