Question: I want it to perform good enough as a multimedia machine. Should I get one that is fan cooled or passively cooled? I think I want one with an HDMI port.

What chipset should I look for?

Answer: 
I agree with an Ion chipset, there's really no competition in the quiet/ITX space. I've got one of [these](http://www.newegg.com/Product/Product.aspx?Item=N82E16813500035&cm_re=zotac-_-13-500-035-_-Product) powering a secondary HTPC for my bedroom. It has a [Celeron 430](http://www.newegg.com/Product/Product.aspx?Item=N82E16819116039) 35W chip in it, and it runs good and quiet. HD playback is no problem, once I got a codec that uses the hardware decoder instead of trying to shove it through the CPU.

My original thought was to go with the Atom 330 version of this board, but what I've seen about it lead me to believe that it would be too sluggish when using the VMC interface. 4 l-cores or not, its still an in-order architecture. If silence is more important to you, there are some Atom Zotac boards that are passively cooled, and have a power brick instead of a normal PSU.
If you want a really small system, look for a machine based on Nvidia ION - basically, it has a Atom CPU and can do 1080p output whilst keeping electricity usage low.

I have not seen many machines recently that are passively cooled, but a lot of the newer Atom boards have low noise fans.