DotOrg, which is run by Public Interest Registry, is quite a different concept to the more popular .com.
While .org can be registered by anyone and used for just about anything, the extension has always had a strong focus on non-commercial users eg. NGOs, foundations, research projects, open source advocacy etc., etc.
Sites using .org include:
Google.org
Apache Software Foundation
Greenpeace
Grassroots.org
You can get more information over on the PIR site.
On a personal note, I am delighted that we are now working with them directly, as their staff are always so incredibly helpful and open. You can follow them on Twitter too.