A distributed system is kinda like computers going social. Instead of one computer having everything and doing everything on its own, a whole bunch of computers connected to a network are involved in the stuff that they do. They mare share resources, processing power and files.
Multiplayer Online Gaming can be an example of a distributed system. And the same can be said about computers running p2p programs like shareaza, limewire... And another great example is Lightwave 3D using its network-sharing feature to allow multiple computers on a network to help do the rendering of the 3D.