FC/NES copiers *don't* backup games so CopyNES is your best option. CopyNES has been recently discontinued however so you're on your own to build one! Other than CopyNES all that's left are DIY projects.
There is currently no accurate way to backup FDS disks! If someone says there is, they're lying.
FDS Loader and PasoDisk are the best options but still not satisfactory for archival. If you're interested in talking about accurately archiving disks, come over to NESDev.
There isn't an affordable copier which can backup games. This is unfortunate because PCE is perhaps the easiest console to backup ever. The best known units for PCE backup are the Magic Super Griffin and MGD2 (with the rare backup board). If you can find Hucard connectors, I can sell a backup device 10-30x cheaper than the copiers mentioned above with better accuracy too!
Mask of Destiny's Transfer tool is your best bet, it's the cheapest and most reliable way to backup games and more. Super Magic Drive or Double Pro Fighter are really the only suitable copiers for dumping and even then they're iffy, all other copiers are pretty horrible at backing up games.
The ROMs must be desoldered or removed from their sockets and read with a device programmer such as the Willem.
The most common copiers such as Game Doctor SFVI/SF7, most Pro Fighters and Wild Cards are the most suitable for backing up standard games. Magicom, Supercoms, Multi Game Doctors, Multi Game Hunter, Super Disk Interceptors etc are not.
Dumping games with special chips requires a SNES expertise (custom dumping algorithm).
Any GB "Linker" made by Bung or UFO or Gamars or EMS or CCL or practically anyone else will backup most GB/GBC games. In the case of old test carts with EPROM, you could always use a device programmer instead.
The few games using memory mappers not supported by linkers need to be dumped by people with GB expertise (custom dumping algorithm).
SMSReader is pretty much the only solution. In the case of GG/MK3, you'll need to salvage a connector from a dead GG/MK3 and build a SMSreader with the GG/MK3 connector using the proper GG/MK3 pinout.
Almost any GBA "linker" is suitable for backing up games, with a PassMe like device, DS games too.
Super Acan/Jaguar/Game.com/Colecovision/Intellivision/Lynx/VCS/other shit:
There are currently no backup devices available but I'd consider making one :)
CDROM consoles like PCE/TG16 CD, Mega/Sega CD, 3DO, Jag CD, CD32, PSX, Sat etc etc:
Use this tutorial:
If you don't know what you're doing, there's no way in hell you can make a perfect image of a disc with CD audio. Most people that even think they're experts don't have a clue about accurately archiving CD audio games.
DVD games (PS2):
Just use CDRWin or ISObuster Trial and make sure you don't use any error correction! Don't use Nero, it will erroneously give you good dumps.
-Modchip or other way to load .DOL executables
-Load GCOS with backup feature
-Connect BBA to PC via crossover cable or a switch
-Follow GCOS' instructions...
GC NR discs:
Do the same thing as GC but use a NR cube. If you have a modchip installed in a retail cube and you want to backup a bunch of NR discs, just move the NR disc assembly over to the retail cube.
-Buy a specific DVDROM drive
-Flash new hacked firmware
-Get disc imaging software
More details elsewhere.
If there's interest in a multi console cartridge backup device (most of the above mentioned consoles) I'd be happy to work with some members to come up with something feasible and make it happen. I know how to interface with all the consoles mentioned in this post and could probably add support for just about any other console, where I'd need help is in finding the necessary connectors.