I used to simply suggest buying the largest PC enclosure possible, and going with cumbersom and expensive extra accessories... Not any more...
There are very few, if any, PC enclosures specifically designed for quiet computing. We are finally starting to see more and more cases now, however. One of the newest, and best looking cases is the Antec Sonata. In my opinion, this is the perfect case to start with when building a silent computer. If you already have a suitable tower to use, there are things you can buy that can make things quieter; I'll get to those later on.
It looks rather unassuming, I agree. But it is built with us in mind; hard drive brackets contain rubber grommets to dampen vibration sounds. And rather than have a pair of loud 80mm fans on the back, a single large fan is used for exhaust. Coupled with one of the silent power supplies we talked about on the silent power supply page, nothing is going to stop you. NOTHING. You are going to have a VERY quiet PC.
The Antec Sonata II takes the original design one step further, by employing an integrated fan duct to direct air flow. It also comes equipped with a 'lower noise' 450W power supply. There are plenty of fan ducts, so you can customize the installation to make the Sonata II as quiet as you want, or beef it up for better cooling performance. It's really quite a versatile case, but I'd personally replace the power supply and exhaust fan.
The Antec P180 is a more performance-oriented case than the Sonata series. However, it does have some special features which help it keep things cool. It can easily be converted to a 'quiet' case by using other silent parts we recommend on this site. Something like this is recommended for those with really hot, high end components in their system, where it would be nearly impossible to go fanless.
It has 2 chambers, separating the PSU from the rest of the system. This isolates heat from the system and lowers system noise. Also, it has a 3 layer side panel, which helps dampen the sound of any fans you might have installed.
Home Theatre Cases
If you're looking to build a silent home theatre PC, my single recommendation is the Silverstone LC03. It supports FULL SIZE ATX motherboards, and is well spaced-out to allow for ample cooling. I have a fanless HTPC system in one of these, and it's very reliable. Even with a fanless cpu (Zalman CNPS7700), fanless video card (Gigabyte X700), fanless power supply (Silverstone ST30NF), and no exhaust fans, my system has been pumping along without a hitch.
Silent Case Fans
Most of the case fans we used to recommend are no longer being made, or hard to find. I'll do some more research, and update the article later.