I may not have phrased that right. I was just wondering, how does a computer organize data going in and out from processor registers? I mean, since multiple programs would of course be accessing it at once, I was just wondering. Also, what exactly is this directed by? Sorry for so many questions, but ooonneee more. Where does the computer store register data for another program that was previously in a register, if another program accesses the same register?
Multiple programs do not access a single core's register at the same time. That access is carefully coordinated by the operating system. You will need to take an operating systems class or read a good book on the subject to get the details. But basically the OS is responsible for storing a program's CPU state (including registers), switching between active programs, and restoring that state. That can happen hundreds of times per second.
A class or a book dedicated to the subject is a better suggestion as they would contain a lot more detail but since they are a large investment in time and\or money I thought I'd link you to a sample first.