The Internet is very simple to describe, but it involves us going into relational theory: for a graph is a relation on a finite set.
We call the nodes, hosts. We assume a set of hosts can communicate with each other, and that one of them is elected as the gateway. The gateway resides on two or more networks, but is not the gateway for all.
Thus when a gateway receives a message, it might be for any number of hosts to which it is directly connected; if not, it needs to choose between the other gateways to which it is connected.
Hosts are all capable of being gateways: that of being connected to more than one network is the best definition of a gateway. With a VPN, then, our host is its own gateway. Traffic for the private network is allocated to the VPN interface. Sending to a host on the VPN involves encapsulating the message and sending it to the VPN server (through the other network, obviously).