Red5 is probably the more well known open source streaming server capable of live streaming. That said, the prices of commercial live streaming servers has fallen dramatically. It's been a while since I tinkered with Red5, but I found it an overly "immature" project.
- Adobe Flash Media Server: This server set the standard. Used to be prohibitively expensive, but is cheap these days
- Wowza: Java based, so it runs on just about anything. Their support forum has tutorials and people there who will help
Both Adobe and Wowza allow limited servers (10 streams) for free.
I just recently whipped up a demonstration setup with Wowza and QuickTime Broadcaster. Everything went really well. I was impressed with Wowza.
If Open Source is really, really important to you, then another server you might want to check out is the Darwin Streaming Server, although that is not Flash video oriented.