higan is the new name for bsnes. As the author of bsnes added more and more features including emulation cores for consoles other than the Super Nintendo, the name bsnes didn't fit well anymore. "higan" apparently means "Hero of Fire".
higan has currently the most accurate (perfect) emulation of a Super Nintendo. If you are looking for an emulator that can emulate just about any Super Nintendo game flawlessly, this is it. Unfortunately the price for accuracy is higher CPU cost. If your PC isn't very powerful you may need to use something else.
higan supports the following consoles, but out of this list its Super Nintendo (Super Famicom) and Gameboy Advance support is probably the best, you may want to check if there are better emulators available for the other consoles in this list.
Nintendo Famicom (NES)
Nintendo Super Famicom (Super Game Boy, BS-X Satellaview, Sufami Turbo)
Nintendo Game Boy/Color
Nintendo Game Boy Advance
Sega Master System/Game Gear
Sega Mega Drive
NEC PC Engine
A particular quirk of higan is that its interface is maybe not the friendliest. One thing for example you may need to do before playing is configuring the controls as by default no keymap is set up. Even for playing fullscreen no hotkey is assigned by default.
The way higan handles ROMs is also a little different: it'll copy every ROM you load with it to an "Emulation" folder in your home folder (this is configurable) and put each game into it's own folder. Things like save states and save games are then stored alongside in that folder. The author of higan calls this system "Game Paks" and a full explanation is available on the official site.
higan is also available for OS X and Linux, but you have to compile it yourself. If you have questions or comments there is an official message board for higan.