Which suffix?

In theory, should generally use .com for commercial domains, .org for organizations, .net for communications networks. Other popular suffixes are .info, .tv (for hosting video).

If you're a non-profit, you should probably register and promote the .org version of your name, but also register the .com version to prevent someone else from using it.

Ownership

You should own your own domain name. Many times your webmaster will own it. You should be the registrar, administrative, and billing contact. Allow the webmaster to be the technical contact.

http://www.keytlaw.com/urls/whoowns.htm