user105: 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?

user91: According to DNS Lookup wwww.bitcoin.org is (today) just another another name for vhost.sourceforge.net so it is run by the sourceforge guys. As the DNS Authority for the www subdomain is bitcoin.org you might be interested in that as well.
According to WHOIS bitcoin.org is run by: 
Domain ID:D153621148-LROR
Domain Name:BITCOIN.ORG
Created On:18-Aug-2008 13:19:55 UTC
Last Updated On:18-May-2011 04:47:13 UTC
Expiration Date:18-Aug-2019 13:19:55 UTC
Sponsoring Registrar:eNom, Inc. (R39-LROR)
Status:OK
Registrant ID:02e87e389a273cab
Registrant Name:Louhi  Net Oy
Registrant Organization:Louhi Net Oy
Registrant City:Helsinki
Registrant Country:FI
Registrant Email:hostmaster@XXXXXXXX
Admin ID:02e87e389a273cab
Admin Name:Louhi  Net Oy
Admin Organization:Louhi Net Oy
Admin City:Helsinki
Admin State/Province:
Admin Country:FI
Admin Email:hostmaster@XXXXXXXX
Tech ID:02e87e389a273cab
Tech Name:Louhi  Net Oy
Tech Organization:Louhi Net Oy
Tech City:Helsinki
Tech Email:hostmaster@XXXXXXXX
Name Server:DNS1.LOUHI.NET
Name Server:DNS2.LOUHI.NET
Name Server:DNS3.LOUHI.NET
DNSSEC:Unsigned

Although it does not look very trustworthy this is a common hosting setup. It may differ from NIC to NIC but usually the real "owner" or his responsible person is mentioned as Admin.

user63: As pointed out by theymos on Bitcoin Talk, both bitcoin.org and bitcointalk.org are owned by Sirius.

user105: 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.

105: <p>The original client that can be downloaded from <a href="http://www.bitcoin.org">www.bitcoin.org</a> is often referred to as the "official client". Similarly many regard <a href="http://www.bitcoin.org">www.bitcoin.org</a> as the "official homepage" of the project. So, who runs it? Is it the core developers? Some Bitcoin association? Or just a private person?</p>
 91: <p>According to DNS Lookup <code>wwww.bitcoin.org</code> is (today) just another another name for <code>vhost.sourceforge.net</code> so it is run by the sourceforge guys. As the DNS Authority for the www subdomain is bitcoin.org you might be interested in that as well.</p>

<p>According to WHOIS bitcoin.org is run by: </p>

<pre><code>Domain ID:D153621148-LROR
Domain Name:BITCOIN.ORG
Created On:18-Aug-2008 13:19:55 UTC
Last Updated On:18-May-2011 04:47:13 UTC
Expiration Date:18-Aug-2019 13:19:55 UTC
Sponsoring Registrar:eNom, Inc. (R39-LROR)
Status:OK
Registrant ID:02e87e389a273cab
Registrant Name:Louhi  Net Oy
Registrant Organization:Louhi Net Oy
Registrant City:Helsinki
Registrant Country:FI
Registrant Email:hostmaster@XXXXXXXX
Admin ID:02e87e389a273cab
Admin Name:Louhi  Net Oy
Admin Organization:Louhi Net Oy
Admin City:Helsinki
Admin State/Province:
Admin Country:FI
Admin Email:hostmaster@XXXXXXXX
Tech ID:02e87e389a273cab
Tech Name:Louhi  Net Oy
Tech Organization:Louhi Net Oy
Tech City:Helsinki
Tech Email:hostmaster@XXXXXXXX
Name Server:DNS1.LOUHI.NET
Name Server:DNS2.LOUHI.NET
Name Server:DNS3.LOUHI.NET
DNSSEC:Unsigned
</code></pre>

<p>Although it does not look very trustworthy this is a common hosting setup. It may differ from NIC to NIC but usually the <a href="https://en.bitcoin.it/wiki/Sirius" rel="nofollow">real "owner"</a> or his responsible person is mentioned as Admin.</p>
 63: <p><a href="https://bitcointalk.org/index.php?topic=33393.msg420189#msg420189">As pointed out by theymos</a> on Bitcoin Talk, both bitcoin.org and bitcointalk.org are owned by <a href="https://en.bitcoin.it/wiki/Sirius">Sirius</a>.</p>
 105: <p>Collecting some of the publicly available information that I found.</p>

<p>Martti Malmi (<a href="https://en.bitcoin.it/wiki/People">aka Sirius</a>), a <a href="http://www.bluewhitereflections.net/index.php/2011/05/the-scary-and-fascinating-currency-of-the-future">student of Aalto University</a> 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 <a href="http://bitcointalk.org/?topic=16.0">here</a> and he was the one making the first commits to <a href="http://bitcoin.svn.sourceforge.net/viewvc/bitcoin/">the Sourceforge repository</a> back in August 2009. He was removed from the list of project developers on bitcoin.org in June 2011.</p>

<p>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) <a href="https://bitcointalk.org/index.php?topic=21549.msg271339#msg271339">have access</a> to the bitcoin.org website.</p>

<hr>

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