Emulators are programs that allow software to be run on computer hardware that they were not originally intended for. The most well-known arcade emulator is MAME (Multiple Arcade Machine Emulator).
There are numerous emulators out there. easyEmu has tutorials for many of these emulators.
To make it somewhat easier, we'll break them down into smaller categories:
Emulators require rom image files to work. The rom files contain game data such as program code, grahics and sound. Be aware that many rom images contain copyrighted material. See this Wikipedia section for more information about the legal status of ROM images. Note that it is not allowed to post ROM image requests on the BYOAC forum. Many other arcade hobby related sites follow this rule too.