I wrote this up for bored_to_life, but figured I'd repost it for anyone looking for advice. If you're reading this way after I've actually posted it, know that some of the technologies I say you should be sure to get, may no longer be the best.
A typical shopping list looks like this: Case and Power Supply Motherboard Processor Memory Hard Drive Video Card
These days Sound Cards and Network Cards are usually optional, because most new motherboards come with these onboard. Lots of them even have 7.1 sound.
Usually good brands: Cases: Antec Motherboards: Abit, ASUS, MSI Memory: Corsair, Crucial, Kingston Hard Drives: Seagate, Western Digital Video Cards: eVGA, XFX, ASUS, MSI, Leadtek, PNY, BFG Tech
You can ask someone else and they will probably have a similar list just missing a few and with a few extra. These are the names that stick out in my mind. YMMV.
Case and Power Supply
The biggest thing I think about when cases come up, is that power supplies that come with cases usually suck. Antec seems to be an exception to this, but be sure to search around the net for reviews of the particular model you find that you may like. I bought an Antec P150 because I was making an effort to build a silent PC. The power supply that comes with it is modular (you can attach only the cables you need) and has really done a good job.
Another thing is, you don't need a 1000 watt power supply unless you are running four video cards and twenty hard drives. Seriously, don't worry about getting a big power supply unless you actually start having power issues. I'm running a GeForce 7900GT and three harddrives on a 430 watt and having no issues.
First decide between AMD and Intel. Right now it seems like Intel is actually the biggest bang for the buck, and being an AMD fanboi, that's hard for me to admit. When shopping for processors I recommending checking Tom's CPU Charts. It hasn't been updated lately, but it will give you a good idea of how everything compares. The kind of motherboard you get will depend on which brand of processor you choose.
First decide between nVidia and ATI. I personally recommend nVidia, but when I was building a PC a few months back, I wasn't really sure why. Just a personal preference I guess. Check out Tom's VGA Charts when shopping for a video card. It will help you pick a chipset, and it looks like it was recently updated. You can't just go by the amount of memory a video card has. After you've picked a chipset, you'll have to compare stuff like clock speeds and pixel pipelines on the cards. You'll want to be sure and get a PCI Express card.
I recommend getting a motherboard with an nVidia North Bridge chipset. Also, make sure it has SATA 3.0GB (Same as SATA2) for your hard drives, and PCI Express for your video card. Search for reviews (beyond just NewEgg) of the boards you think you may get.
There's really not much to picking a hard drive. You should make sure it is SATA 3.0GB (Same as SATA2). Then you can compare cache (more is better) and write and seek times (lower is better).
Read up on "Dual Channel" and you'll see why memory is usually paired in matching sizes. Really 2GB (two 1GB sticks) should be plenty to start with. Check what type the motherboard you picked supports and go from there. You don't have to buy the memory pair kits, but I would stick to buying the same brand. If you can find a case where buying two single sticks is cheaper than buying a pair kit, go for it.
I bought a $30 NEC DVD drive from NewEgg and it has worked just fine. I've done plenty of CDs, and the few DVDs I've done I had no problem with. I really don't know much about these though.
Check what kind of warranties your parts will come with. If you're having trouble deciding between a couple of parts, warranty can be the deciding factor. Be sure to keep up with all the receipts and paperwork you'll need if a part goes out. Usually the warranty you have an individual part will be longer than the warranty someone gets buying a computer from Dell or something.