Apple is the company. "A Mac" refers to a computer than runs any Mac OS (Macintosh System 7, Mac OS 8, Mac OS 9, Mac OS X, etc.). "The Mac" refers to the OS itself (more commonly, people will just say "OS X"). The grammar has variations. Some people will refer to the Mac OS simply as "Mac", the way you'd simply call Microsoft's OS "Windows". There's certainly logic to that, but...I dunno, it just doesn't sound right (to me), so I say "the Mac [OS]".
Mac is not a company or sub-division or anything like that. And it is never ever an acronym, so anyone calling it MAC (all caps) is just yelling stupidly.