Your biggest restriction would be the $1000 including monitor, but I managed to do the same for a bit more ($1300) two years ago (which included flat-panel monitor and 5.1 surround), so I can offer some tips.
If you build it yourself, you can usually get a level up in performance for the same cost as a pre-built, but look at places like the slickdeals site that was linked for pre-built computers with rebates. You could pay something like $1250 and get $250 back in a few months, and get a computer that would have otherwise cost more to build on your own, since the manufacturers generally get discounts on hardware for buying lots of the same components in bulk, so the savings get (partially) passed on. But, what it basically comes down to is flexibility and reliability. Pre-builts have limited flexibility, but are generally more reliable because all the components are guaranteed to work together flawlessly. Building your own gives you ultimate flexibility, but you can run into problems with parts working (e.g. some CPUs or memory will pass factory tests but fail when put into an actual computer).
Now, for specifics on building your own:
CPU, it's sort of a tossup. AMD offers slightly better performance/dollar, but IMO nothing can quite beat an Intel chip on an Intel board at the moment. If you want to invest in a 64-bit processor, definitely go with AMD. Intel is lagging behind severely in that department. But there aren't many applications out now that will make use of the 64-bit capability (and I believe the version of Windows that supports it is still beta?), but it's sort of "the next thing". I'll leave the pros/cons of 64 bit to the people that actually have those processors.
Memory, like was already said, 512 is a minimum now (especially for Windows). Mine has 1GB, which I'll be upgrading soon to either 2 or 3. I would say it's worth the cost for what you're doing because you WILL be multitasking. I'm not saying this from a computer-knowledge standpoint, I'm saying it from a university student standpoint. Plus, if you're going to be gaming on it, 1GB would probably be the absolute minimum. Besides, you can get that much for under $100 now anyway.
Motherboard, depends on your processor. If you're getting Intel, definitely go with an Intel board. Otherwise, look for the amount and type of memory supported, PCI slots (does it support newer PCI-express?), the AGP slot (video card might need 8x, etc.), any on-board stuff you may want to use (I would only use sound and NIC). Also, you'll want probably four USB ports. If Firewire is included too, that's just gravy.
Graphics Card, I am a convert to nVidia. I've just had far too many problems with ATI drivers being utter crap. And I would say don't bother with the 256MB cards, most of that extra memory doesn't even end up being used. I believe the top-of-the-line right now is the 6800, but I would just check out the prices and get a mid-range one (probably around 6200 by now). It should last for a few years and be at least tolerable for the newest resource-hog game.
OS. Before you do anything, check with your school. When I first started school, Win2kPro was offered for FREE to students. You just had to know where to find it
I can now get WinXPPro for free, as well. I would also recommend you get the ISO for a Linux LiveCD (Knoppix, Ubuntu, Kubuntu) to at least try it out. Linux is free, has geek cred, and you can guarantee that most people won't want to mess with your computer because it "looks weird". Only downside is most games are Windows.
Drives. Go with a DVD-R (or DVD+R, whatever). They're cheap now, they can burn CDs as well. Might want to possibly get a cheap CD reader as well, since burners tend to have a lower MTBF (mean time before failure... basically how many hours on average it's gonna work before blowing up). Since I've bought the DVD burner, I've gone through about 50 DVDs, most handed out to other people. YMMV, but it's still a very good thing to have.
Hard Drives, look around a bit. Most of the 60-80GB models are priced at a bit over 1$US per GB, but the bigger 160-200GB drives can be had for around 0.50$US/GB. Big things here are interface (don't want to get slower ATA100 when you can have faster ATA133), rotation speed (7200 rpm better than 5400 rpm... 10k rpm is usually SCSI only, and very expensive). Also look for MTBF if it's listed. Bigger numbers here = more reliable.
Case/Power supply: DO NOT GO CHEAP ON THIS. It will fail on you. And, because of Murphy's Law, it will be at the exact moment when it will do the most damage for you. I know people who went all-out, spent $2000 on their system, and got a crappy $20 case/PSU combo. One of my friends got a cheap case that didn't even have the panel to hold the systemboard aligned correctly, so it just hung by two screws. I even knew a guy that tried to build his own case from scrap metal. Spend some time and find a good quality case that allows for ventilation, easy access, and solid construction. You might also want to invest a little in a battery backup. I got one rated for 425W for about 4 minutes, which is enough to check email one more time, save everything, twiddle thumbs for 40 seconds, and shut down with no problem.
Monitor: while it's good to get whatever fits within budget, do not underestimate the value of desk real-estate in college. Right now I have the flap extender thing up on my desk, and my dresser up next to it for more space, and there still isn't enough for me. And I have a flat panel. I guess it sort of depends on your vigelence with organization skills.
And some stuff you're missing:
Input devices (keyboard, mouse): You'll want these. Cheap route is beige clunker keyboard and ball mouse. I got a wireless keyboard and USB optical mouse. Just depends on what you'll be doing.
Speakers. If you're gonna be gaming, it's blasphemy to do it with no sound. Plus you'll probably be doing stuff like music and watching videos/movies. I got an Altec Lansing 5.1 for $100, and the card to support it for another $40. You can get a decent 2.1 computer speaker set that will work with onboard sound for fairly cheap.
Printer. You will have papers. Your professors probably will expect hard copies. It's inconvenient, but it's a fact of life. Check if any of the provided labs provide printing services first, but a cheap inkjet is always a nice thing to have.
And, one more thing: try looking into Apple. I recently got a PowerBook, and have been doing a lot of work with it. I am now addicted to XCode, and catch myself using Expose shortcuts on my PC. The mini and ibook have been upgraded, with the mini being $499 sans monitor, and the iBook being $999. Worth looking into.