Q:

Who runs www.bitcoin.org? 

The original client that can be downloaded from www.bitcoin.org is often referred to as the "official client".  Similarly many regard www.bitcoin.org as the "official homepage" of the project.  So, who runs it?  Is it the core developers?  Some Bitcoin association?  Or just a private person? 

A:

Collecting some of the publicly available information that I found. 
Martti Malmi (aka Sirius), a student of Aalto University in Finland, was one of the core developers early on in the project.  Satoshi attributed some of the new features of version 0.2 to him here and he was the one making the first commits to the Sourceforge repository back in August 2009.  He was removed from the list of project developers on bitcoin.org in June 2011. 
As Andrew points out in his answer, both bitcoin.org and bitcointalk.org is owned by Martti.  He is the administrator of the bitcointalk.org forums, still active there, and could most likely be contacted using the PM feature on the forums.  Both Martti and Gavin Andresen (lead developer) have access to the bitcoin.org website. 

Edit September 25th, 2011: The code for the bitcoin.org website is now hosted under the Bitcoin project on GitHub.  This means that you can post issues/suggestions there or make a pull request if you wanna add or change something. 

A:

As pointed out by theymos on Bitcoin Talk, both bitcoin.org and bitcointalk.org are owned by Sirius.