You have a lot of different options for older systems. In general:
NES: FCEUX (Primary), Nestopia (Backup)
FCEUX is constantly updated and has SVN builds every few days. Nestopia is considered the most "accurate" NES emulator, so it lacks a few high end options FCEUX offers.
Sega Master System/Game Gear: GensGS (Primary), Kega Fusion (Secondary)
I personally prefer GensGS, for reasons about to be stated below.
Sage Genesis/Sega 32x/Sega CD: GensGS (Primary), Kega Fusion (Secondary)
The reason I prefer GensGS here is because it supports Sega CD Savestates, even if it isn't implemented 100% correct yet. Kega Fusion lacks Sega CD Savestates as of the last time I used it. That being said, the Gens-derived emulators (Gens, Gens32, GensGS, etc) all have issues with optical drives, due to the emulator being based on Win9x code.
SNES: snes9x (Primary), ZSNES (Secondary)
I prefered ZSNES for years over snes9x, as it has some very nice features (including settings that eliminated internal SNES slowdown). But its DOS based root are showing its age; custom GUIs aren't that good an idea anymore. Both are solid choices, but snes9x has aged better.
PSX: PCSX-Reloaded (Primary), ePSXe (Secondary)
PCSX-Reloaded passed ePSXe during its long stay at version 1.7. Extra options (native Gameshark support, BIOS emulation, for instance) eventually made their way into ePSXe 1.8, but I still prefer PCSX-Reloaded. In both cases, I recommend Petes OGL2 GPU Plugin and the Eternal SPU Plugin.
Note some games are REALLY picky with the CD-ROM plugin (CAPCOM, I'm talking to you here), as most don't read audio tracks properly in some games (Resident Evil/Dino Crisis being the two most known offenders). I'll post the CD-ROM plugin I use when I get home (I *think* its Moodys, but I forget offhand...)
N64: Project64 (Primary), 1964 (Primary)
Emulator/Plugin choice is VERY game dependent. Some games WILL NOT RUN or WILL NOT RUN WELL.
PS2: PCSX2
Only choice. Just remember the GSDX > ZeroGX in all cases, and you'll be fine.
Gamecube/Wii: Dolphin
Only choice again. Some games WILL NOT RUN. Dolphin also has noticeable speed issues, and the HLE audio engine is crap.
Arcade: MAME (Primary)
Barring a non-MAME supported game (anything in 3d mostly), use MAME. Other games are a case-by-case basis on which emulator (if any) to use.