Peer-to-peer (P2P) applications and their filesharing networks -- like BitTorrent and Gnutella -- are a great way to mutually share your files with others, but as with so many good things, they can also lead to trouble. For example, you could download copyright-protected material or as a nasty bit of malware (viruses, spyware and trojans) without meaning to, or you might leave yourself open to sharing personal information you'd rather not have in the hands of strangers. You could also put others at risk for copyright infringement if they were to download one of your files.
Entertainment owners such as the RIAA, NBC Universal and the MPAA actively survey computers connected to the Internet in search of violations of copyright law, sometimes using the same file-sharing software as the entertainment traders do. This automated process is often performed by other organizations, such as IP-Echelon (used by HBO among others). They will check for files stored in a "sharable" folder, looking for those whose distribution to others would be considered unauthorized.
Copyright is the legal protection of intellectual property, in whatever medium, that is provided for by the laws of the United States to the owners of copyright. Types of works covered include, but are not limited to: literary, dramatic, musical, artistic, pictorial, graphic, film, multi-media and digital works, and software. It also includes the digital transmission and subsequent use of these works, such as sharing your favorite song with others via a P2P network.
In other words, you may legally own the song or video, but under copyright law you would can not share it without incurring a copyright violation. Brown's Copyright Infringement Policy explains about the illegal use of filesharing programs and University policies and procedures for handling violations.
Read Use P2P Filesharing Software Safely & Legally to learn how to use legitimate P2P applications, remove P2P apps, and find alternatives to illegal filesharing.