Mac OS
Mac OS is the operating system used in Macintosh computers since their introduction in 1984. Some appliances (notably the iPhone and iPad) also run derivative versions of Mac OS. The current version is Mac OS X 10.6 (often referred to as simply OSX), which is derived from NeXTSTEP, and has a Mach microkernel running a BSD compatibility layer. Most applications are written in Objective-C using the Cocoa framework.
"Mac" is a term which describes the broader class of Apple computer hardware and their Macintosh operating systems.
Older versions
Pre-X versions of Mac OS were written in-house at Apple and were based in part on the slightly earlier Lisa Office System. "Classic" Mac OS is still in use on many older Macs, especially 68k Macs (Macs with a Motorola 680x0 processor), which are unable to run versions newer than 8.1. (In fact, many older Macs can't even run OS 8 at all.) Depending on the specific model, PowerPC Macs support anywhere from System 7.1.2 up to OS X 10.5. (OS X 10.6 dropped support for PowerPC Macs.)