ROM
ROM, short for Read Only Memory, is a kind of computer memory that cannot be written to (at least after it has been programmed). Super Nintendo cartridges contain a form of ROM memory in chips.
ROM images are computer files containing a dump of the data from ROM memory, for example (and usually exactly what is meant when the term is used in the ROM hacking community), in a video game cartridge. ISO files and BIOS dumps (when the BIOS is on a ROM chip, like on a Game Boy Advance) could also be considered ROM images.
Emulation
The machine code stored in an SNES ROM is written for another CPU architecture than that of a modern personal computer, meaning that one needs a program that interprets the ROM's programming. Such a program is called an emulator. Without an emulator (or an actual SNES), the ROM is useless. After all, in the same way a Sega Genesis cannot play Super Nintendo games, a modern personal computer cannot directly execute SNES ROM files, either. Even an Apple IIGS, an ancient PC of the 80's, with its 65c816 CPU could not run SNES ROMs natively due to other architectural differences (eg. video, sound, storage).
In simpler terms, the computer speaks one "language" while the ROM is written in another. An emulator can then be described as a translator between these two "languages", allowing the ROMs to be played on a computer.
Legality
ROMs are typically illegal to distribute on the Internet and illegal to own. For that reason ROM patchers and patches exists, such as BPS or IPS so they can be distributed without having copyrighted content.