Each iPod MUST be associated with a Computer with iTunes installed. For ease of explanation we'll call this the "Home" iTunes.
When you first setup and configure your iPod you will do this on the Home iTunes and that "locks" the iPod to that computer. Every time you plug the iPod into the computer it should synch some or all of the music/video and data from iTunes with the iPod.
All "purchased" music, bought through iTunes Store can be freely copied over and back between the iPod and the Home iTunes without any problem. All music on your iPod should be already backed up on your iTunes.
HOWEVER when you plug your iPod into another computer, it will ask you 1st if you want to erase it and link it to the this new computer and iTunes. If you say NO, then it will load the ipod and you can see and play all the music on the iPod but iTunes WILL NOT allow you to copy or move any music or videos from the iPod to iTunes. This is Apple's attempt to stop people sharing music and videos illegally or easily .... but as always there are ways around the problem.
There are several reasons why you may want to copy music from your iPod to a version of iTunes what is not your "Home" iTunes. A common problem is when the computer crashes or fails and you have to start with a new PC .. then your only copy of music may be on the iPod. Apple cannot help so we get a 3rd party application to "restore" your music to the computer. There are FREE and commercial programs available. Personally I use a free and simple program called .. Yamipod
http://www.yamipod.com/main/modules/homeRemember that music bought on iTunes has rights-management included and therefore you need to "authorise" the new copy iTunes to play the music.