You can use any type of iPod with either an Apple or a PC. You must install and use iTunes as it's default music program. iTunes is really easy to use and free to install. The iPod will come with iTunes software or you can download it from the Apple site.
The iPod shuffle is a type of iPod but it does not have a display and you cannot pick what order your songs are played in.
Every other iPod has a screen and you can pick the order your songs are played in, create playlists (think mixed tape).
When it comes down to it an iPod is a regular MP3 player. Some other MP3 players out there are the Nomad and the RIO. MP3 players all have the same basic functions. They play music. The iPod is trendy and supposedly the easiest to use. I have an iPod because I've had one since the beginning.
If your daughter is going to be running or exercising with it you may want to get her a flash drive based player (shuffle, nano from Apple.. don't know the non-iPod ones) instead of a hard drive player (regular iPods) because the movement when running is bad for the hard drive.