You can emulate most platforms from before and after the Atari 2600 up to the Nintendo Wii, depending on the power of your computer and the quality of the emulator. For instance, 2600 and PlayStation emulation is great and do not require strong computers by any means these days, but 3DO, Jaguar and Saturn emulation can be less than ideal. Obviously newer platforms, like the Dreamcast and Wii, require far more powerful computers.
Emulators themselves are legal, but ROMs of commercial software are not. Sort of. I've always been under the impression that if you own a game, then getting a ROM of it should be legal (you have the game/cart in-hand already). I could be wrong, but I don't really care--I emulate regardless. If you are concerned about it, download homebrew ROMs that are offered for free. That is legal provided the original author meant them to be available in that manner.
So, you want to set up an emulator for first-time use? Start with something simple, like the NES or Atari VCS/2600. Generally you just install the emulator, then place your ROM files in whatever folder you want. From inside the emulator, you can load the ROM, and the game will start as if you had a cartridge inserted into the original console and flipped on the power switch. Later systems, including the PlayStation, Neo-Geo CD, and more, all require extra BIOS images, so if you are confused already, don't start with them first.
To get an emulator of your choosing, find a site based around emulation. It's old, but I still revert back to Zophar's Domain by habit (zophar.net, I think?). Anyways, a quick plug on a search engine will help you find loads of other sites that host emulators.
If you want to emulate ON a console, you can do that too. Emulators exist for the Dreamcast, the XBOX, the Wii, and more. Also, you can get emulators on portable devices like the PSP, N-Gage, Zodiac, and iPhone. Some of these devices need to be cracked, hacked or jailbroken though (respectively) before you can install the emulators.
The Wii is the most ideal if you want to run emulators on a console and not a computer. It's the cheapest, it's the easiest to hack, and supposedly the easiest to use/most user-friendly after everything is loaded. The XBOX isn't as powerful as the Wii, but it's still a monster, doing everything well up through the PlayStation, even the N64 (compatibility is what hinders N64 emulation on it though). I believe if you have custom firmware on your PS3, you can emulate on that, too. It's by far the most powerful of the three named, but I'm not sure how easy it is to do, not to mention I'm not sure I'd want to do it to that since they are still relatively expensive consoles.
Edited by Austin, Mon Aug 29, 2011 8:14 PM.