What part of the operating system allows users to communicate with the computer system?

Answer 1

The answer is the "shell." The shell takes two forms:

Command line interface (Like cmd.exe or BASH.), what is often just called a "shell" in many circles. You enter a command, arguments, or even construct entire strings of commands or scripts and the computer will, provided they are valid.

The other is the graphical user interface, examples are Aero, Explorer, Plasma Workspaces, twm, Quartz. Usually if you use your mouse and click on icons or menus to launch your applications rather than invocation from the keyboard.

Further, when you use application software you are also leveraging communication with the operating system. Any time your application needs to make any use of the hardware for any reason, it has to negotiate with the operating system to get services from the hardware. In modern operating systems, applications have zero control over the hardware, they usually have to tell the kernel when they need something outside of the CPU.