Proxying provides Internet access to a single host, or a very small number of hosts, while appearing to provide access to all of your hosts. The hosts that have access act as proxies for the machines that don't, doing what these machines want done.
A proxy server for a particular protocol or set of protocols runs on a dual-homed host or a bastion host: some host that the user can talk to, which can, in turn, talk to the outside world. The user's client program talks to this proxy server instead of directly to the "real" server out on the Internet. The proxy server evaluates requests from the client and decides which to pass on and which to disregard. If a request is approved, the proxy server talks to the real server on behalf of the client and proceeds to relay requests from the client to the real server, and to relay the real server's answers back to the client.
As far as the user is concerned, talking to the proxy server is just like talking directly to the real server. As far as the real server is concerned, it's talking to a user on the host that is running the proxy server; it doesn't know that the user is really somewhere else.
Since the proxy server is the only machine that speaks to the outside world, it's the only machine that needs a valid IP address. This makes proxying an easy way for sites to economize on address space. Network address translation can also be used (by itself or in conjunction with proxying) to achieve this end.
Proxying doesn't ...