In a laptop, this is rarely possible, but in regards to desktops, see below.

There are VERY few motherboards that have the CPU (processor) soldered on the board. I'm fairly certain 100% of all new(er) computers (Read 7 years and newer) have removable CPUs.

I'm not sure why you would want to remove the CPU though, especially if it's an older computer as you'll be very hard-pressed to find any upgrades for it.

In regards to the operating system...you can't have one without a motherboard or CPU.

The motherboard is the body of the computer (nervous system, etc.) and the CPU is the brain. That's about the best metaphor I can put it in.

I hope this helps.