The popular peer-to-peer file sharing application founded in 1999 was Napster. Created by Shawn Fanning and Sean Parker, Napster launched on June 1, 1999, and made it easy for users to search for and exchange MP3 music files directly with one another.
Napster was technically a hybrid system: a central server indexed users' shared files and helped them find one another, while the files themselves were transferred between users' computers. This design made music sharing dramatically more convenient than older methods such as FTP servers, bulletin boards, or mailing physical media.
Napster became enormously influential but quickly faced copyright lawsuits, including a major case brought by the Recording Industry Association of America. Courts eventually ordered the original service to stop facilitating unauthorized sharing, and it shut down in 2001. Kazaa, LimeWire, and BitTorrent appeared later or used different approaches, so they are not the canonical answer to this 1999 question.