I built this little thing for the sole purpose of playing GOG.com's back catalogue of games (with a bit of emulation on the side). It's not very powerful, but it doesn't need to be; the games I like are fairly light on resources, and my current backlog is enough to last me a good 4-5 years (I can never resist GOG.com's relentless sales...). It's my first build, and probably far from perfect, but I think it turned out all right. Let me go through my component selection:
CPU: I originally planned to use a Pentium G2120, which would have been perfectly adequate. But the supplier (dabs.com) messed up the order, and I had to cancel and look elsewhere... at which point I realised that (including next day delivery charges and random price cuts) a Core i3 would only cost ~£20 more. I couldn't resist.
CPU Cooler: I debated using the stock cooler, but kept reading bad things about the push pin mounting system. As a newb, I felt safer going aftermarket. The NH-L9i certainly lives up to its reviews (excellent quality, silent fan) and it's the perfect height for the case, but the reviewers who said it was 'easy to mount' deserve a poke in the eye. I felt like a blind man trying to dock with an orbiting space station. Where did the thermal paste end up? Your guess is as good as mine... But at least with screws I know it's securely fixed (it doesn't bend/stress the motherboard, either, which is nice). Under full load (Intel Burn Test) in a warm room, the CPU never exceeds 61C, so I'm happy with the cooler's performance.
Motherboard: I wanted something cheaper, but (a) the bargain boards might have needed a BIOS update for Ivy Bridge (didn't want to risk it) and (b) the economical ASRock models looked pretty good, but I noticed a high DOA rate among the customer reviews (didn't want to risk it). The ASUS is fine, though (decent build quality, sturdy PCB, easy to use BIOS).
RAM: I just chose the cheapest DDR3-1600 dual channel kit (although I do trust Corsair). I need no more than 4GB, as I'm running Windows XP (32bit) for maximum compatibility with older games (I know it's EOL in a few months, but I'm not going online with this machine. DRM-free software = no internet connection required).
HDD: I just chose the cheapest '3.5 inch 7200RPM 16MB cache' drive that didn't get mixed reviews...
GPU: I only game on a 1360x768 display (an LG telly), as I've found this to be the best resolution for all the old Infinity Engine RPGs (and their contemporaries). I also like seeing pixels. Not sure why. But anyway, an HD 7750 is fine for 720p-ish gaming, and it's also one of the most energy efficient cards on the planet (and Sapphire seem to make the best edition, provided you pick the GDDR5 version). It runs cool (60C under full load with FurMark) and I can't hear it over the case fans.
Case: I have little space, and the Elite 120 was the only small case that fit my budget. I was pleasantly surprised by its quality. It's very sturdy, and the layout makes assembly quite easy (things only get fiddly once the PSU goes in). Airflow is a bit dodgy, but I added a second 120 mm fan to the drive cage (note that this has to go in before the motherboard...), and installed the PSU fan-side down to exhaust hot air. Seems to work reasonably well. Cable management took ages, but wasn't too onerous (there are loads of holes and loops for cable ties). You can also easily remove unnecessary front panel connector cables, which helps considerably. The case fans seem to be the only source of noise in the build, but it's not too bad (about as loud as an Xbox 360, but 'softer').
PSU: I just went for the cheapest, good branded semi-modular model (I wanted to minimise excess cabling). The whole PC draws ~100W at full load, so I don't need anything special...
I haven't had much time to play with it yet, but the few random games that I tried (Driftmoon, Trine, Stacking, Two Worlds II, The Witcher) all ran flawlessly at max settings. I also had a quick dabble with the Dolphin Emulator, and had no problem running Wii games at 720p. Overall, I'm very happy with the build.
Oh, and I'd like to thank pcpartpicker.com, without whom I'd still be adrift on the web, clicking through incompatible parts like a little lost soul...