Ok, here is a build. When sticking to a tight budget, there will always be compromises. I believe such compromises can and should be targeted carefully, so they don't prevent the system from becoming useless in a matter of months.
ASRock B75M LGA 1155 Intel B75 HDMI SATA 6Gb/s USB 3.0 Micro ATX Intel Motherboard
Model #:B75M
Item #:N82E16813157313
$69.99 -$5.00 Instant $64.99
I've been happy with ASRock. This board has modern interfaces and features, and should remain capable for years to come.
PowerColor AX7770 1GBD5-DH Radeon HD 7770 GHz Edition 1GB 128-bit GDDR5 PCI Express 3.0 x16 HDCP Ready CrossFireX Support ...
Model #:AX7770 1GBD5-DH
Item #:N82E16814131477
Mail in Rebate
$119.99 $119.99
This card is a competent gaming card, particularly at the resolution of the selected monitor. At the time you consider upgrading your monitor, then you may also want to upgrade this card, but even then you may not have to if every setting need not be on UltraSuperMaxOhWOW.
Rosewill R101-P-BK 120mm Fan MicroATX Mid Tower Computer Case
Model #:R101-P-BK
Item #:N82E16811147112
$29.99 $29.99
I have built in this case. It is reasonably sturdy for a cheap case, and has plenty of room for the selected components. Its fan is quiet but moves enough air, especially considering that this initial configuration won't generate a lot of heat. Should you upgrade the processor and/or graphics card in the future, an additional $10 fan in the front should take care of any extra heat.
LITE-ON DVD Burner - Bulk Black SATA Model iHAS124-04 - OEM
Model #:iHAS124-04
Item #:N82E16827106289
$17.99 $17.99
One optical drive is much like another.
Intel Core i3-2120 Sandy Bridge 3.3GHz LGA 1155 65W Dual-Core Desktop Processor Intel HD Graphics 2000 BX80623I32120
Model #:BX80623I32120
Item #:N82E16819115077
$124.99 $124.99
I was actually surprised to be able to fit this into the build, thinking I'd have to go for a Pentium G850 instead. If you'd prefer a stronger graphics card ($50 will let you take a small step up to a HD6870 or GTX560), you can get the Pentium instead, and still outperform an AMD 965 in most games.
Microsoft Windows 7 Home Premium SP1 64-bit - OEM
Model #:GFC-02050
Item #:N82E16832116986
$99.99 $99.99
Not much to say here.
Gateway HX2001L bmd Black 20" 5ms Widescreen LED Monitor
Model #:ET.DW1HP.001
Item #:N82E16824113030
$139.99 -$50.00 Instant $89.99
You could spend another $20 for a 1920x1080 monitor, but it wouldn't be LED, wouldn't have speakers, and would be closer to the comfortable limit of the chosen graphics card. IMHO this monitor fits this build; simple as that.
Logitech B120 Black 3 Buttons Tilt Wheel USB or PS/2 Wired Optical Mouse
Model #:910-001802
Item #:N82E16826104379
$14.99 $14.99
Bottom-dollar rodents are frustrating. You won't feel like hurling this one across the room because it can't track, and the tilt wheel gives you some extra options in games.
Microsoft Keyboard 200 6JH-00026 White USB Wired Standard Keyboard
Model #:6JH-00026
Item #:N82E16823109234
$13.99 -$5.00 Instant $8.99
On a budget, this one should do. I prefer light keyboards because they are more readily visible in any lighting conditions; you may want a black one. Feel free to replace this at your discretion, since keyboards are such a personal choice item.
G.SKILL NS 4GB (2 x 2GB) 240-Pin DDR3 SDRAM DDR3 1333 (PC3 10666) Desktop Memory Model F3-10666CL9D-4GBNS
Model #:F3-10666CL9D-4GBNS
Item #:N82E16820231396
$19.99 $19.99
Four GB of RAM is still sufficient, but this probably IS the first place I'd toss another $15, to get a 2x4GB kit. The mobo only has two slots, so upgrading RAM means throwing away or selling this. Still, like I said, on a budget this is enough.
Western Digital Caviar Blue WD5000AAKX 500GB 7200 RPM SATA 6.0Gb/s 3.5" Internal Hard Drive -Bare Drive
Model #:WD5000AAKX
Item #:N82E16822136769
$79.99 -$10.00 Instant $69.99
It pains me to choose a drive with only a two year warranty, but these seem to hold up well enough. For another $40, I'd get a 1TB WD Black, which still has a five year warranty.
Antec VP-450 450W ATX 12V v2.3 Power Supply
Model #:VP-450
Item #:N82E16817371045
$49.99 -$15.00 Instant $34.99
Efficient enough for 80+, it is only disqualified for not having an active PFC circuit (it needs a little voltage switch). When you upgrade your graphics card to a HD7850, you won't need to replace this PSU.
Subtotal: $696.88
There's shipping on some items, but there's also a rebate in there, and perhaps a discount code.
When you can, you'll want to add decent speakers. You may eventually want a larger monitor (and then possibly a stronger graphics card), but I believe this system is well balanced for its budget. It is a good place to start, and has the ability to grow, so it won't feel like a waste next year.