Basically people had ripped their CDs and put the music into MP3 format on their computers. Napster searched the client's computer for mp3 files and created an mp3 list which was uploaded to Napster's server and made available across the entire network whenever that client's computer was logged in. Hence, using a Napster's central database list, people were able to connect to each other's computers and share music files which (at least at one point) had been legally purchased.
The question was, according to the Home Audio Recording Act (
http://en.wikipedia.org/wiki/Audio_Home_Recording_Act), how much of these file transfers were considered protected personal backups of owned recordings or protected transfers between friends/family, and how much of it was copyright infringing distribution of intellectual and physical property.
Because Napster ran a central database of clients' mp3 lists (as opposed to a true peer-to-peer network where there is no central database), the RIAA went after Napster and won.
I'm not saying it's right what Napster did, but I am saying it's wrong how the RIAA decided to prosecute them and other bigtime mp3 distributors. The record labels chose to shut down a totally new and innovative avenue of distribution that was (in hindsight) the wave of the future instead of embracing it.
They're paying for it now, though! Record labels are becoming a thing of the past with digital distribution, since basically their main purposes were advance funding, marketing, and physical album distribution. Indie bands can take care of the marketing and distribution side though things such as iTunes and Myspace, and not have to spend millions producing a clean, professional recording. Maybe artists will actually start to make money off their recordings immediately, as opposed to only after the record label had recouped all its' advances.