How can the Operating System run on the same chip it is supposed to be managing? From my readings about Operating Systems (reading the basic material on Wikipedia, tech sites, etc) I've learned that the Operating System is a program that allows programs and applications to interact with the hardware in an efficient and safe way.

However I'm confused about how the Operating System oversees the computer's operation when it itself needs to be operated.

What do I mean? Well, the way I would imagine an Operating System to work, is that on a computer, there would be two CPUs. One that runs the OS all the time, and another that the OS uses to run the computer.
However, it turns out that the OS is running on the same CPU that the other processes are. This is like a manager having to work on the same production line as his employees, and only gets to use the power tools when another employee is done with them. He would not be a very effective manager, since he wouldn't have the ability to issue orders if his employee is even slightly undisciplined.

So how can it be that the OS only runs part of the time on the same CPU that has to be shared between all the other processes? How does this end up working out?
In their most primitive form, operating systems rely on processes being well-behaved and handing control back to the operating system from time-to-time. If a process is not well-behaved, then indeed it can happen that the operating system freezes.

Most CPU's however, have interrupts: an interrupt pauses the current process (preemption) and hands control back to the operating system. You can schedule interrupts to happen at specific intervals, or be triggered externally (for instance pushing the reset button).

Basically, the CPU switches back and forth between running the OS and running processes at a blazingly fast speed. Furthermore, CPU's have "user mode" and "kernel mode". Processes run in user mode and the OS runs in kernel mode. Running in user mode prevents a process from executing restricted commands (such as IO). If a process wants to write to the disk, it has to ask the OS to do so. The OS switches to kernel mode and does the write on behalf of the process (after making sure the process has the required permissions).