Final Fantasy VI Emulation
Sorry, we're too clean-livin' to help you find illegal ROM downloads. You should be able to find a version somewhere, though. The normal FF3us ROM is probably what you want, for now.
i90east's Expert Version 2 Hack
CoN member i90east has hacked the original FF3 ROM to add difficulty and a new necessity for strategy into the game, along with a number of other changes as detailed on his Web site. The file here is a patch to the original game, and must be used with a patching tool such as the one found in SNESTool to be combined with the original FF3us. If you don't know how, try asking in the forums.
CoN member Rubicant has hacked the original FF3us to restore old Japanese names to the characters and items wherever possible, and has added the CzarDragon to the active enemy roster (In the Ancient Castle, WoR). This file is a patch to the original game, and so requires a tool like SNESTool to patch together. If you don't know how, try asking in the forums.
Windows, DOS, and Linux
The two great emulators for DOS/Windows/Linux are Snes9x and ZSNES. Most people prefer ZSNES, but they are fairly equal at present. Some computers prefer one, some prefer the other. Snes9x offers smooth OpenGL and 3dfx Glide output methods if you have compatible hardware, but the same effect occurs in ZSNES with some newer video cards.
Try them both, and make your choice. Since the emulators will be updated periodically, we link only to their homepages here.
Other Operating Systems
Snes9x, as linked above, has been ported to a number of different operating systems. Check there to see if a version exists for your system.