The most obvious consumers of processing power are the applications and programs you want the computer to run for you. From a spreadsheet to a database, applications are the reason you have a computer.
A single-CPU system can only do one thing at any given time. Therefore, if your application is running, everything else on the system is not. And the opposite is, of course, true -- if something other than your application is running, then your application is doing nothing.
But how is it that many different applications can seemingly run at once under a modern operating system? The answer is that these are multitasking operating systems. In other words, they create the illusion that many different things are going on simultaneously when in fact that is not possible. The trick is to give each process a fraction of a second's worth of time running on the CPU before giving the CPU to another process for the next fraction of a second. If these context switches happen frequently enough, the illusion of multiple applications running simultaneously is achieved.
Of course, applications do other things than manipulate data using the CPU. They may wait for user input as well as performing I/O to devices such as disk drives and graphics displays. When these events take place, the application no longer needs the CPU. At these times, the CPU can be used for other processes running other applications without slowing the waiting application at all.
In addition, the CPU can be used by another consumer of processing power: the operating system itself.