https://computer.howstuffworks.com/kazaa.htm
KaZaA didn't store the files itself so it was thought they wouldn't be possible to shutdown. From the site above:
"While Kazaa claims to be "completely legal," there are those who disagree: The free-to-download blue files are controlled by Kazaa users and include copyrighted content."
"Later that year, Kazaa was sued again, this time in the United States by the Recording Industry Association of America (RIAA) and the Motion Picture Association (MPAA). As of February 2005, the decision in that suit is still pending."
I remember they started suing individual users at that time... I found an article explaining that:
https://www.videoproc.com/resource/what-happened-to-kazaa.ht...
"In September 2003, the RIAA filed lawsuits against over 250 individuals, accusing them of illegally distributing about 1,000 copyright music files each, using P2P networks. RIAA sought an average compensation of $3,000 per case."
The result of the first case:
"In July 2006, the MGM Studios, Inc. v. Grokster, Ltd. caused Sharman to settle for $100 million, the amount to compensate the loss of four major music labels – EMI, Sony BMG, Universal Music, and Warner Music. The company also agreed to pay an undisclosed amount to the studios in the industry."
It's unclear exactly how Kazaa got down, the article concludes with "In August 2012, the Kazaa website was no longer active."... "the rise of legal streaming services such as iTunes, Spotify, and Netflix further compounded Kazaa's demise.".
Looks like the music industry managed to scare people away from pirating instead of actually succeeding in bringing them down directly, which is more or less what I remember.
Gnutella brought peer-to-peer searches. Basically it used a flood-fill algorithm: your search would be broadcast to all connected peers, which would broadcast it to all peers that hadn't seen it yet, until somebody responded with the file and their IP and you could download directly from them. Interestingly Ethereum uses basically the same algorithm for block distribution, with some optimizations that were first published by RTM, who was one of the founders of YCombinator.
Kazaa's innovation was to split the peer space into "ordinary nodes" and "superpeers", with the observation that not all bandwidth links were equal. It would enlist hosts on high-bandwidth connections to form quasi-centralized indexing nodes to organize the network topology for all the low-bandwidth consumer nodes. It's a similar principle to how the Lightning Network works for Bitcoin, or how L2s on Ethereum operate. This also made it easier to shutdown than Gnutella though, because being a superpeer made you a legal target for the RIAA.
Kazaa and ed2k were distributed. I think ed2k is still viable.