Compatible with most games, even many ROM hacks that make use of emulator quirks.
higan (or, as it was known before it expanded its scope to more systems, bsnes) is the most accurate SNES emulator out there, but newcomers who care more about playing games rather than 100% accurate emulation should stick with the most recent version of the forked bsnes core from the same author.
Incompatible with ROM hacks made to take advantage of emulator quirks, much like real hardware.
Should play all commercially released games without trouble, assuming you have the power.