Wine (known recursively as Wine Is Not an Emulator) is an open-source compatibility layer for Unix-like systems (e.g. Linux, Mac OS X) to run programs originally created for Microsoft Windows. Originally created by Alexandre Julliard, it is now sponsored by corporate entity CodeWeavers, who use it as a base to create an enhanced, proprietary version, known as CrossOver.
It can run older Windows only games and Windows only emulators.
Wine is especially useful for older Windows games, as compatibility with these is typically higher than if the same game were run on modern Windows versions.
Wine can also run any Windows program, including emulators. Compatibility may not be perfect however, and there can be various glitches introduced.
Binaries (via repositories) for Linux distributions
With some programs, official Windows DLL files are required. Most of these are available to download via the (usually) included script, winetricks. Compatibility lists for most programs can be found on the AppDB.