I do not know which of the above are current, and this site has a much more comprehensive list of quiet computer makers. Bear in mind that not all machines from these companies are necessarily quiet; only the ones designed to be so.

SID FutureClient is a fanless system. Good performance, stylish design, and apparently a reasonable price tag.
ECS introduced a small, cheap (albeit slow) silent computer called the iNote. It normally uses a small DiskOnChip (solid state hard drive) that's compatible with Linux and WinCE. You can either add an internal hard drive or a PCI card. It also comes with USB, so you can add external hard drives, additional network cards (if you want a router), etc. It comes with built-in networking, modem, VGA, TV-out, etc. so it is usable on its own. Personally, I'm drooling over this thing, and it looks like it'll be dirt cheap, so I may buy one myself once it has resellers in the US.
The Microspace-PC is small and fanless. Made in Europe.
CalmPC makes a case based on liquid cooling that virtually eliminates noise. There are no fans or motors, although some reviewers have reported a very faint dribbling noise at about the level of an analog wrist watch. It can cool the graphics card, processor and chipset. Runs about $200 with shipping.
ArmSystems makes the Stealth series of quiet PCs.
Captech builds PCs that run at only 16dB.
A neat little fanless thin client, mostly targetted at retail use in music stores.
Fujitsu has long been one of the frontrunners in the drive for silent PCs.
AST launched a line of quiet PCs a few years back.
NEC builds several quiter models
Compaq has been working with Seagate to develop quieter hard drives for their PC line
Many of Dell's PCs are designed to be moderately quiet
To save power, some notebook computers are engineered not to use fans. Coupled with a quiet notebook hard drive, these can make very nice, quiet PCs. There's a large premium, of course.