Peer-to-peer software allows users to share files with each other over a huge, decentralized network.  There is no big server but rather every user's computer is like a mini server in the network.  You can allow other users to download any of your files like music, movies, and programs and they can share any files they want to with you.  
             
Some popular P2P programs:

Kazaa
Morpheus
Limewire
Ares

Using peer-to-peer software to download music is not a good idea. It's illegal and there are viruses and other junk that can end up on your computer when using p2p.

Illegal Downloads:
A majority of the files shared on p2p networks are illegal to download. You may think it's great that you can get your favorite songs for free but you can be caught and fined, or put in jail, for downloading illegally.  It's the same as swiping CDs from a music store and it is a good way to ensure that your favorite up-and-coming artists won't have the money or drive to continue making great music.

Viruses:
Some users on p2p services like to hide viruses in files that they share.  If you use peer-to-peer you are very likely to get spyware and viruses on you computer.  Also, some of these services come preloaded with annoying pop-ups and adware that you can't get rid of without disabling the service.

Be sure what you're getting:
Some users will download a song and rename it for whatever reason. The song is then passed to another user who knows the title is incorrect and so renames it again.  Sometimes you will get music that you think is from your favorite artists but it turns out to be something completely different; or you may find a great song and want to get more songs from the same artist but are unable to find them because your download was labeled incorrectly.

Full of Junk:
P2P users share lots of objectionable material.  These networks are overloaded with adult material, violent videos, and explicit music without the warning labels or effective filtering system that you get with legal downloading software.  This material is nearly impossible to avoid when using p2p.  Some users will even label questionable materials incorrectly, purposely, so you will download it and use it thinking it is something benign.