Peer-to-peer systems are unique in that everything about them is truly distributed. One of the interesting things about peer-to-peer systems is that they scale instantly. The more peers join a system, the more computing bandwidth is available. As a result, a peer-to-peer system can quickly transfer data and cool down rapidly where as a client/server architecture can take a while to cool down. P2P networks are a perfect medium for sharing files because many machines can share the burden of getting a file to a user. This can enable a user to truly leverage all their available bandwidth because they are receiving data from multiple nodes.
Distributed architectures are becoming crucial to the success of large companies on the web. Content distribution networks are used to spread out content and data so that users might be able to request it from a source closer to their location. It would be an interesting study to look at putting P2P features into a web browser so that static content could be requested from your closest neighbor. It would also be interesting to look at a P2P type active network for pushing dynamic content out onto the network.