A virtual private network (VPN) is a computer network in which some of the links between nodes are carried by open connections or virtual circuits in some larger networks (such as the Internet), as opposed to running across a single private network.
E.g. A home computer can be set up so that it appears to be on a work network, even though all data has to travel over the internet.
A virtual private network is a way of linking computers into a network with the same privacy and security as a LAN but by using public internet connections to do so. A VPN is a much cheaper way to connect company networks than a dedicated leased line.