Red5 is a popular program written in Java that lets you stream audio/video files (FLV/F4V/MP4/3GP/MP3/F4A/M4A/AAC), record streams (FLV), and supports various protocols (RTMP/RTMPT/RTMPS/RTMPE).
To get started, we will spin up Ubuntu 12.10 x64 cloud server:

Install Red5
apt-get install -y red5-server
This will install Red5 to /usr/share/red5
To start red5:
service red5-server start
Your server will now listen on these default ports:
RTMP: 1935
Debug proxy: 1936
HTTP servlet: 5080
RTMPT: 8088
At this point you are all done installing Red5, and now you need to add your Red5 application to /usr/share/red5/webapps
You can develop your own application using Eclipse IDE. There are many commercial applications available for Red5, as well as free applications.
There is also an online Red5 App Generator available here.
It will let you create RTMP streaming application:

Here is an example of how to install a free Red5 Chat application:
cd /usr/share/red5/webapps wget "http://downloads.sourceforge.net/project/freered5chat/red5chat_v2.2_Freedevelopment.net.zip" unzip red5chat_v2.2_Freedevelopment.net.zip mv red5chat_v2.2_Freedevelopment.net/* .
Restart Red5:
/etc/init.d/red5-server restart
Navigate over to your cloud server's IP port 5080 (http://198.199.91.87:5080/flash/ in our case):

And you are all done!