Well, if you want a new desktop, you should definitely build your own. Assuming you're marginally computer literate, although, if you want to run emulators, especially more modern ones, you pretty much have to be, so we're going to make that assumption (note: building your own computer means you won't get customer support from a PC manufacturer, so you'll have to do most of your own troubleshooting, but you should have a warranty on individual parts, if you buy them new).
Now we're done with disclaimers, I'll post some links that'll get the basics done:
http://www.pcmag.com/article2/0,2817,2347249,00.asp
http://www.extremetech.com/article2/0,2845,2346481,00.asp
Those are pretty good for walking you through building a PC, but they're both really low end (400 and 500 dollars, respectively, and the first uses integrated graphics). My recommendation would be to spring for a better (or best) motherboard, since it's the hardest piece to upgrade, and if you buy one that is better than the rest of your PC, you can upgrade the more important parts later, according to your needs. I'd recommend using those guides as a template for building your PC (and it has pictures!), and picking the parts that are either more cost effective or flat out more powerful, if cost isn't an issue for you. For modern PC games, a good video card and a lot of RAM is more important than a CPU, but I'm willing to take the word of the person in this topic who said emulators need better CPUs (that seems to make sense to me). If you want to play PC games and use a Wii emulator, you'll want a decent graphics card, but if not you could probably skimp a little on what graphics card you use, though I'm not sure if you'll want to use onboard graphics with your mobo. I can't really give more specific instructions without knowing your needs better, but, in general, computers are most cost-efficient around a thousand bucks.
Oh, and make sure your powersupply can handle all the components you put in. This may be helpful, but you'll want to double check the numbers:
http://extreme.outervision.com/psucalculatorlite.jsp
