Most exploits rely on creating input to the software that causes it behave in an unexpected way, causing it to execute instructions to hardware not originally intended. To fill that in with an appropriate level of detail would probably take half a bachelor's degree worth of lectures in computer architecture, software, hardware, etc. But you could also start with this link, and every time you encountered a term you don't know what it is, click the clink to its own Wikipedia article, or search Google for that term, and gain as deep a level of understanding as you feel like.