Firstly you'll save money (aside from some killer sales) by building yourself rather than buying pre built. There's also a lot of things you may or may not need. For example, sound cards are not a necessity at all as mobos these days come with high def 7.1 audio. Unless you're an audiophile and have some decent speakers or really good quality headphones, mobo sound is fine.
IMO go to Tom's Hardware forums, there's a ton of people posting builds all the time and a lot of very knowledgeable people so it's a great place to learn.
AMD CPUs are great for budget builds. Intel CPUs are faster but mostly unessary for gaming. If you can afford an i5 750 build tho, they're really good. Otherwise an AMD Phenom X4 955 or 965 is a great way to go.
Also, I recommend not getting a sound card for now. You probably won't care for on. I got an AuzenTech Bravura 7.1 (~$100) and I love it with my 7.2 speaker set up, but for 2.1 or headphones it's not all that necessary as mobos come with high def audio onboard anyway.
Video card is very important. Right now the best bang-for-you-buck high performing GPU is the Nvidia GTX460 768mb. The 1gb version is about 5-8% faster, but costs a little more too and that 1gb is only really needed for higher resolution settings. Still, it's a good way to go.
From ATI, the 5850 is a great GPU and is one notch higher than the 460 on the heirarchy. IMO, if you pick a motherboard with SLI support get the 460 so that down the road you can easily stick another 460 in it and get superb performance. Otherwise you need either single card Nvidia or you can crossfire ATI. That said, if you play at, say, 1280x1020 or something then a 5770 is a fine GPU and much cheaper. It's not going to max games out at 1920x1080 but it's a decent performer nonetheless. Nvidia GTS 260 I believe would be similar, but then you won't have DX11.
Make sure you buy Windows 7 64bit
Get 4GB RAM. Speed/latency isn't a big deal. If you were thinking of overclocking the CPU, then get 1600mhz RAM. If not, 1333mhz RAM.
PSU is important. Get a quality one. My preffered brand is Corsair. They offer great quality PSUs with a ton of amperage. I believe Antec has stepped up their PSUs lately, so that's fine too.
For a case, just get a mid size tower, as long as it can support ATX motherboards. A really good budget case is the Antec 300 - but it really comes down to your taste. Buy one you like.
CPU cooler is a good idea but only required for overclocking. My suggestion, to keep it cheap, is the Hyper 212+ which is great for it's cost.
Anyway, we can throw numbers and product names at you all day, but you really need to do some research so your not dissapointed in the end. And http://tomshardware.com is a really great place to start. Once you start thinking about and picking some items, google reviews and you'll find other good sources - overclockers.net, silentpcreview.com, frostytech.com, guru3d.com etc etc. There's plenty.
Start by matching a CPU and GPU - a higher end GPU will do better with a faster CPU. Then based on that you can get the mobo, RAM, PSU... etc.
NOTE: http://www.tomshardware.com/reviews/balanced-gaming-pc-overclock,2699.html
Part 4 of a series "Building a Balanced Gaming PC". Lots of info to take in but it's good to read through.
EDIT: I'll make this a super long post by adding a $1000 complete PC I put together for another thread. I made it off newegg.ca, so it's under $1000 Canadian. I'd recommend a 1080p monitor but otherwise this PC is very good.
Phenom II x4 965 Black Edition +
BIOSTAR TA890GXE AM3 AMD 890GX SATA 6Gb/s HDMI Micro ATX AMD Motherboard
Combo deal $269.98
http://www.newegg.ca/Product/ComboDealDetails.aspx?ItemList=Combo.456495
CORSAIR 4GB (2 x 2GB) DDR3 1333 RAM $89.99
http://www.newegg.ca/Product/Product.aspx?Item=N82E16820145278&cm_re=dual_channel_ddr3-_-20-145-278-_-Product
Corsair 550VX PSU $99.99
http://www.newegg.ca/Product/Product.aspx?Item=N82E16817139004&Tpk=550vx
Antec 300 case $54.99
http://www.newegg.ca/Product/Product.aspx?Item=N82E16811129066&cm_re=antec_300-_-11-129-066-_-Product
Western Digital 500GB HDD $54.99
http://www.newegg.ca/Product/Product.aspx?Item=N82E16822136497
Philips 19" LCD Monitor 1440x900 $114.99
http://www.newegg.ca/Product/Product.aspx?Item=N82E16824133039
Cyber Acoustics 2.1 speakers $39.99
http://www.newegg.ca/Product/Product.aspx?Item=N82E16836150086
ASUS DirectCU TOP GTX 460 768mb $214.49
http://www.newegg.ca/Product/Product.aspx?Item=N82E16814121390&cm_re=gtx_460-_-14-121-390-_-Product
Cheap keyboard $12.49
http://www.newegg.ca/Product/Product.aspx?Item=N82E16823107128
Cheap laser mouse $19.99
http://www.newegg.ca/Product/Product.aspx?Item=N82E16826193039
Sony Optiarc Black 24X DVD+R Lightscribe $23.99
http://www.newegg.ca/Product/Product.aspx?Item=N82E16827118040
Grand total: $995.89 (Canadian)