MAME is an acronym for Multiple Arcade Machine Emulator. The intent of this emulator is preservation; the ability to play games is considered a “side-effect.” MAME supports thousands of classic arcade games, although not all of them are playable.
For legal reasons, this site does not contain nor link to ROMs. But MAME is an exception, as the “ROM site” in question is none other than the Internet Archive (archive.org). You can find the game ROMs, BIOS ROMs and all other files needed to play games in the MESS and MAME collection on their site as both direct downloads and torrent files. Be aware that a complete collection can take a lot of disk space (over 50GB, and you need double that to extract the archive that holds all the ROMs). After downloading you can place game and BIOS ROMs in the “roms” subfolder inside the MAME folder.
MAME handles ROMs a little different in that you do not have to unzip every individual game. MAME can open them as ZIP files. The reason for this is that arcade ROMs tend to consist of multiple files.