This thread is loosely based off of the ZSNES help thread, I'm just improving on it a bit.
Ask any questions regrading any emulators.
Along with that, here is a list of a few of the more well-known Emulators and their sites that may also help you with what you need.
Note: I'll update this list as time goes on. If you feel I missed anything, feel free to post a link to it and I'll update this list. Just please make sure that the site DOES NOT contain any links to a ROM.
NES
FCE Ultra X - All the major FCE Ultra offshoots merged together. Recommended for Tool Assisted Speedrunning.
NEStopia - A easy to use, customisable emulator that supports just about everything. Recommended!
SNES
ZSNES - Probably the most popular emulator.
SNES9x - Not quite as popular, but still useful. Sound emulation's iffy with some games, though.
SNESGT - A Japanese emulator in English. Has good sound emulation!
bSNES - Extremely accurate to the original machine, so system requirements are very high. Crappy computers need not apply!
N64
Project64 - The most popular emulator. Development has stagnated and the latest version is only available to donators or people who are very good at finding things.
Mupen64 - Development has stopped, and the latest spinoff (Mupen64plus) is current OSX/Linux only, but this might be useful to someone.
Next-Gen
Dolphin - A Gamecube AND Wii emulator? That WORKS? It's a Christmas Miracle!
Portables
VisualBoyAdvance-M - THE GBA emulator.
NO$GBA - A great emulator for GBA and DS.