macOS 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, named iOS. The current version is macOS Sierra (10.12), which is derived from NeXTSTEP, and has a Mach-based microkernel running a BSD compatibility layer. Most applications are written in Objective-C and Swift using the Cocoa framework.
Versions of macOS from 10.0 to 10.11 were branded as Mac OS X, and the current release is still unofficially referred to as such. Previous versions were simply called Mac OS, while the oldest versions were referred to as System followed by the version number, e.g. "System 7".
"Mac" is a term which describes the broader class of Apple computer hardware and their Macintosh operating systems.
Pre-X versions of Mac OS (Mac OS 9 and lower) 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 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.)