In my recent experience and in overly simple terms; if you are looking for the best bang for your buck (budget) go with AMD, if performance (and bragging rights) is more important snag a nice Intel i7.
Keep in mind going with an i7 may drive up the price of other components as well since the mobos and flashy DDR3 memory is usually a quite a bit more pricey. Also, if you are looking at it from a gaming perspective, as I recall, not too many games at the moment are optimized to take full advantage of the quad cores but, this obviously is/will change and having a nice quad core may make your new system more future proof and upgradeable.
A few random tips that you most likely already know about building a system in no particular order:
1. Get a nice power supply! A cheapy PSU usually is going to be less efficient, most likely no where near what it is really rated in power, and quite possibly could fry your entire system if it goes out. Bad news all around.
2. Make sure your mobo has all the connections you need, particularly if the case you get has plugs for firewire, eSata, etc. (seems obvious but still happens to the best (and worst

) of us)
3. Check that the fancy memory you get is good to go with your motherboard... sucks not getting the best clock speeds or having an unstable system just because your motherboard doesn't support a certain brand.
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Like others have already said, the video card is a huge choice but luckily Nvidia and ATI seem honestly equivalent to me at the moment... Nvidia used to have an edge but ATI has made up a lot of ground. Really can't go too wrong with either brand as long as you get a nice one. ATI may have a slight advantage innovation wise (GDDR5, 40nm) but, Nvidia has equally well performing cards and has a reputation for a bit more stable drivers.
All waaaaay more information than you asked for but, figured I'd share what I learned from when I built a nice budget gaming system a few months back. Have funny putting your new system together! I sure enjoyed building mine.

EDIT:
P.S. I second any recommendation for Windows 7, it has impressed me.