I always go with the Oracle/Sun java and try to not use the OpenJDK version. I usually run into incompatibilities, and I end up having to install it anyway.
You can't just yum install java
you have to go to Oracle's website and download Java/JDK directly from there. I'd recommend getting the latest version if you can.
The naming of Java has always been a bit of a mess, I'd suggest downloading the version from the URL above, it's the Java SE version. Make sure you select the .rpm
version and also make sure you select the appropriate version for your system's architecture (32-bit or 64-bit).

It's pretty straightforward. Once you download it you'll need to execute the download, which should begin the installation using the contained .rpm
files.
I like using this method since it allows me to manage the Oracle JDK just like any other RPM on my system, so it's easier to uninstall it and also you can then have multiple versions installed along side one another.
Running red5
As @terdon has mentioned in his answer, red5
is pretty straightforward to download and execute. I just confirmed that it works fine with the OpenJDK version of Java on my Fedora 19 system, I would expect it to work just fine on CentOS 6.x without issue as well. I also tried it with Oracle's JDK so either installation of Java should suffice here.
I'll just add the following extra bits to the directions in @terdon's answer. I'd suggest changing directories to the directory that red5
was untarred into like so:
$ cd /path/to/red5
$ ./red5.sh
After it starts up you can see what ports it's listening on using this command:
$ netstat -tapn | grep java
(Not all processes could be identified, non-owned process info
will not be shown, you would have to be root to see it all.)
tcp6 0 0 :::1935 :::* LISTEN 9419/java
tcp6 0 0 :::47663 :::* LISTEN 9419/java
tcp6 0 0 :::9999 :::* LISTEN 9419/java
tcp6 0 0 :::5080 :::* LISTEN 9419/java
After starting the server you can confirm that it's starting by navigating to the URL http://localhost:5080
. You'll see a video load on a page like this:
