To listen to Toonami Radio, you need a program capable of playing streaming MP3s. We suggest you get the full versions of these programs (the 'lite' versions are smaller and quicker to download, but lack the ability to listen to music online). Don't worry, these are all free.

Winamp Standard
This is our favorite Windows player. Its small and you can add on new graphic interface called "skins". Nice playlist, easy buttons, and easy to use.
For Windows 95+

iTunes
It encodes, has a playlist, equalizer, visualizations and a lot more.
For Mac OS 9.0+

Real Player Basic
An all around good media player. Plays music, and offers a lot of its own video content. Recommended for use with its plug-ins. No skins feature though.
For Windows, Mac OS, Unix Solaris 2.6+, Red Hat Linux 6.0+

Once you have one of these programs and have them properly installed, copy and paste this address into the program (hold on ctrl button and press L):
http://www.live365.com/play/78885
If you want to listen in another program, copy and paste the address there. If it supports streaming MP3, then it'll work.