ROM, short for Read Only Memory, is a kind of computer memory that cannot be written to (at least after it has been finalized). 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 GameBoy Advance) could also be considered ROM images.
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 it's 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.
ROMs are typically illegal to distribute on the Internet and illegal to own. However, fair use laws and the wishes of the copyright holder can change that. (I.E. Does the copyright holder actually ENCOURAGE ROM hacks? Is it public domain? Was it released under a permissive-enough Creative Commons license?) It is definitely illegal to distribute Super Mario World ROMs on the Internet, so SMW Central will not host them. Users of the site also cannot provide links to sites that are hosting ROMs.