Of all the awesome things you can do with a personal computer these days (let's review: make and enjoy music, make and enjoy video, meet people around the world via the Net, various games, etc.) by far the coolest must be Emulation; or, for you non-technical readers, making your computer pretend that it's another machine. To emulate a machine, your own machine must be many times faster than the machine you're emulating, so you can count Playstation and Xbox (all generations) out. Even the first Playstation can't be emulated on most machines. Same thing with Macs - a small percentage can but most cannot.
This day and age, however, you can emulate a Nintendo Entertainment System (NES - the first Nintendo in America), a Super NES, a Sega Genesis (presumably a Sega Master System as well), and various arcades and other computers from 15 years ago. But this limitation, if you'd call it one, isn't so bad. The current Xbox and Playstation consoles are reverse compatible and will play their old games, or most of them, as I understand it. If you're emulating, you're not doing it to save money or defraud the publishers (whichever side you choose to take), you're doing it to play something that isn't available and in a new way.
Emulation has created a huge stir in the industry. Nintendo is 110% against it as well as any form of enhancing or changing their games (e.g. cheating devices). None of them are really "for" it - either the creators of these classics are out of the industry and don't care, or they're making new games and would rather you fork over your hard earned dollars for those instead. For the most part, game publishers are content to let their older titles be forgotten. Some have made small concessions, either because of emulation as in, in response to it, but a few have done it on their own. For example Nintendo putting all the NES Mario games in one cartridge on the Super NES ("Super Mario All-Stars"). It's too bad they didn't do the same thing with Zelda or some of their other franchises. But now they're putting a lot of their classic games up for paid download on the Wii - provided you can get a Wii (they're not available here) or you have high-speed Internet (not available here).
Wondering how all this works? As I understand it, in Japan where copyright laws are largely ignored, you can get a device classified as a cart copier, which you plug the cartridge into, and it plugs into the console - like a Game Genie - but a wire connects the device to your computer. You can "back up" a game to the computer, and you can play games off the computer via the copier itself, while connected. Then some hacker in America or Australia found out how to take that backup and run it. The first emulators were buggy, designed to run one or two games, and others may or may not work. Now, emulators can fully recreate classic consoles. Two of the emulators I am most familiar with have rather rude names. NESticle plays the NES games, GeneCYST plays the Sega Genesis games - and ZSNES plays the SNES games. There are a few different choices for each platform, and some emulators might do things a little differently, but a good one should be able to play the game.
Of course, emulated classic console games are a little different from the games as you remember them. First, the game is stored in a file. It's much faster to swap games. Your game's file will not get dusty or corroded, and you don't have to worry about blowing on the contacts in the cartridge and in the console. Second, computer monitors and digital TVs have higher resolutions than analog TVs, so picture will be greatly improved. Then there are some perks. A lot of games don't let you save or make you find a "save point" to save. With an emulator you can save ANYWHERE with the touch of a button, and load just as fast. You can load Mario, jump in the air and save, and keep hitting the load key, and have him fall, fall, and fall again. Some games time you, and they keep the timer running even if you pause the game (e.g. you hit Start). You can pause the emulator, however, and this will stop the internal clock. You can literally have your best time in games like Super Metroid (SNES) by pausing the emulator for your breaks. Also, you can cheat! Say you're playing Zelda 3 (SNES) and you want unlimited bombs. You currently have 8, say. So you tell it to search for 8 - you'll get lots of results. Waste a bomb, then tell it to search all the '8' results now for 7. It'll take a few tries, but you'll find your code. Set it to something reasonable (like 5) and it will be frozen at that value.
It sounds almost too good to be true. Well, it's true, but the legal issue is a big grey area. The commonly accepted rule is that you can have a ROM - that's the game backup file - if you own the original cartridge. But it's hard to find ROMs on regular websites, because the game's publishers will go after them for copyright infringement. So determined emulator users turn to filesharing. You download an archive with, say, every NES game ever made (around 100MB). Are you really going to take what you have the cartridges for - maybe what you had back in the day - and delete the rest? Heck no, you're not! (If you say you would, fine, but most people just keep the whole set.) I personally don't feel that there is anything wrong with emulation, in the moral sense. You can't buy most of these games - with rare exception you can't buy any of them for the PC. (Konami released a pack with the three Castlevania games and two Contra games that were on the NES, on the PC - basically a legal emulator/rom pack.) It's just one of those cases where the law doesn't reflect right/wrong, but rather the interests of a company or companies with money to grease the proverbial wheels - the game industry figures if you're playing the classics, you're not paying for the current games. (One would think they would see this as a hint, and look to the classics for inspiration, but only Nintendo seems to do this - hence why I love my Gameboy/DS so much.)
Still weirder, online communities have differing rules governing talk of emulation. Some allow it, some turn a blind eye, and some disallow it entirely. And then you have fence-sitters, like GameFAQs... Couple weeks ago I was talking about a Playstation game, when someone asked me if I'd like to play it on my computer. I knew Playstation emulation wasn't a possibility that I knew of, but I wanted to see where he was going. So I said sure, email me or whatever - next thing I know, my post (not his) gets deleted, where I accepted. So I guess it was a little fishing experiment - "let's see who we can sucker into asking about emulation here" (as opposed to just disallowing talk of it at all). So for a site like that, one really has to be careful and not take the bait. Lesson learned, I suppose.
For legal reasons of my own, I can't tell you where to get ROMs on here. I could probably tell you where to get the emulators, but a quick Google search for the names as I've given them will do that for you. Comment if you have something to add, but if you'd like to ask a question, you ought to do it via email, where I can be more free to answer the question as best I can.
That all said, if it's something you'd like to do, I wish you all the luck in the world. It will take some messing around in the emulator's settings to get it right, but once you do it will be well worth it.
PS - Most emulators default to the keyboard. You can get a gaming controller for your PC. They used to plug into the sound card on this port that was about an inch wide - now they just plug in via USB, same as anything. Most controllers I've seen use the basic Playstation2 layout, which is probably the best controller layout of the consoles. PC controllers have vibration and come wireless, just like the consoles. And as long as Windows recognizes it, the emulator should be able to use it. You can get a cheap controller for around $20 or less. You can even get two and play 2-player games just as you would on the console you're emulating. (Of course, pre-vibration games will not vibrate your controller if it does that. But wireless controllers should be a go.) Oh, and I believe shooting games (e.g. Duck Hunt for the NES) are emulated through the mouse, though I've never tried these.