A computer's central processing unit (CPU) is what gets things done. At its most basic level, a computer program is a complex series of math problems. The CPU is what performs the calculations that solve these problems and gives you the results you expect. Those results might be anything from making a video game character jump over a flaming barrel to running a spell-check algorithm in a word processor.
Every program you run requires some of the CPU's processing power. Each additional program you run simultaneously means the CPU has to work closer to full capacity. Operating systems coordinate with a CPU to make sure everything runs smoothly. The OS might switch the CPU's focus from one program to another as you switch active sessions on your computer. The OS acts like a resource manager and if it works well, you won't even notice that the CPU is switching between jobs at an incredibly rapid pace.
You might not notice if an operating system is working well. That's sort of the point -- the OS handles complex tasks so that you don't have to worry about them. It's only when things go wrong that you realize how important your computer's OS is to accomplishing tasks.
It's easy to notice the cosmetic differences between various operating systems. Unless you're a programmer, you may not notice how those differences extend below the user interface layer of an OS. Ultimately, when you go down deep enough, every OS has to handle the tasks we've talked about so that computers, mobile gadgets, game consoles and other computing devices perform the way we expect.