The ROMs (a.k.a. Roms) are something (A file, but not sure), stored in all game cartridges. They are actually everything that is in the game (Leves, Maps, Characters etc.). Save files are stored in a different part of the game cartriage. When you insert a cartridge into a console (NES, SNES, N64 etc.), the consloe automatically plays the Rom file.
Playing games on a computerEdit
These Roms can actually be extracted, and stored on a computer. That's some kind of illegal, but it's modern. And you don't need to pay about it. You need an Emulator about that.
You can play Roms with an emulator. Like a Gameboy can't play SNES games, and a NES can't play Wii games, a computer can't play any type of a console game. The emulators simply translate the Rom in something that the computer understands. I recommend you to vist www.emulator-zone.com, to download an emulator for every console you want!