Peer-to-Peer (P2P) is not a web service per say, but a distributed application architecture that allows users to share files directly with each other. The architecture was popularized by the music sharing system Napster, originally released in 1999.
Peers are both suppliers and consumers of resources. See Wikipedia.

Digital Media
Monthly users (million)
1.4B  Facebook
1.3B  YouTube
300  P2P - This is a generic term for all P2P
                I couldn't find any site with a name of P2P. See links below.
255  Twitter
250  SoundCloud 
200  Instagram
110  PlayStation
 75  Steam
 75  Pandora Internet Radio
 50  Vimeo
 48   Xbox Live
 40   Spotify
 38  Hulu
 30  Netflix
 20  Amazon Prime

Links:
Peer-to-peer - Wikipedia
P2P Is Still Bigger Than Spotify, Soundcloud, Twitter, Instagram & Pandora... | Digital Music NewsDigital Music News
10 Great Places to Share Your Music Online
Top Bittorrent Sites - The Best File Sharing Websites | about.com
Internet TV
How to Stay Legal When File Sharing (P2P Programs) | about.com
File sharing - Wikipedia
Tru Optik - audience and consumer demand