Mac OS is a popular family of operating systems. It was one of the first systems to Pioneer the Graphical User Interface. It is also one of few platforms to have survived a move to a new processor architecture. It actually survived two; once from Motorola's 68k to IBM's PowerPC, and again to intel.
The "classic" Mac OS is characterised by its lack of the command line, and encompasses the different Mac OS' from 1984 to 2001. It lacks primitive multitasking and protected memory and is composed largely on 68k assembly code.
Unfortunately due to its age, in order to run apps from the old "Classic" System you need an emulator. You can effectively think of the following as the Macintosh equivalent to DOSBox. Three option are available: Sheep Shaver, Basilisk, and Mini vMac. To categorize them in the simplest way possible, Mini vMac is for black and white 16/24-bit 68000 based Macs, Basilisk is for color 32-bit (020, 030 & 040) 68k Macs, Sheep Shaver is for classic PowerPC Macs.
macOS 10 (re-branded from "Mac OS X" in 2012 to "OS X", and later to "macOS") is the latest version of the Mac OS operating system (initially released in 2001), and is also the basis for iOS. It is instead based on NeXTSTEP (which it's self is derived from UNIX).
Wine is one of the easiest and most popular way for Linux users to run programs written for Microsoft Windows. The Wine team has created a port for OS X that is well maintained and in a usable state.
Boot Camp is an official piece of software included with OS X that assists users in installing and running Microsoft Windows in a partition on the hard drive. The most recent version only includes support for Windows 7 and Windows 8.
The Unix Shell "bash" available on OS X is a very useful tool for doing system tasks. If you use OS X as your main operating system, then it is heavily recommended that you learn how to use it.
Google Code University provides a good starting guide.
Rosetta is a software emulator for intel Macs that enables them to run older (OS X native) PowerPC apps. This can be useful for running games that didn't receive a Universal Binary update post 2005, such as Fallout.
Similar to Microsoft's NTVDM for DOS apps, the classic environment was a compatibly layer for OS X that allowed it to run classic apps at native speeds. Note that the Classic environment is NOT an emulator, and as the classic OS is based entirely on PowerPC (and/or 68k) code, it does not function on intel Macs.