You've gotten some pretty good answers, but I'll throw my 2¢ in.
For general information, there are two fairly similar organizations, AirLifeLine and Angel Flight. Both work nationally, but the internal structure is a bit different. You'll tend to find that although both exist throughout the country, one or the other tends to predominate in different geographic areas.
You'll find a central listing of all of the various organizations at the Air Care Alliance home page,
www.aircareall.org.
I use rental aircraft to fly Angel Flight missions. Two concerns with doing that.
First, insurance. These organizations want you to have insurance (obviously). That means either you carry non-owned aircraft insurance yourself or the FBO is one of those that carries insurance that covers the pilot (not merely covers the FBO for what the pilot does). Angel Flight West, for example, will want the coverage verified as part of the pilot orientation process.
Second is the reason that I don't fly as many as I would like. Rental aircraft have maintenance issues all the time. Missions are usually scheduled long enough in advance that reservation is not a problem, but those last-minute maintenance issues can be a problem. Twice in a row, an airplane that I scheduled for an Angel Flight went down for maintenance at the last minute. Fortunately, both times there was an acceptable replacement available.
But it =is= an exceptionally worthwhile thing to do. I have a story on my web site about an Angel Flight mission. It's at
http://www.midlifeflight.com/devyn.htm