Virtual Private Network
In a nutshell a VPN creates a "secure tunnel" between you and another network. Still not making sense??
The general idea is that any traffic sent over the Internet is not secure and must be encrypted in order to be protected. In the case of some business applications (beyond simple email and web surfing) remote users need to be able to access resources stored on the company's secure network (which is shielded from the Internet by a series of firewalls). A VPN takes the traffic from your machine, heavily encrypts it, and then sends it to a machine on the company network which can decrypt it and send it on to whatever computer it is destined for. The same is done in reverse for traffic destined for your machine from the company network.
To the user it looks like their machine is plugged into the network at the office where all resources are readily available, but in reality a virtual secure pipe has been created between your machine and the company network (using the Internet to transport the traffic).
In Ye Olde Days remote users would use a modem to dial into the company network. In most cases the modems on the company side were secure and could give the remote users full access. Now that applications are more bandwidth hungry and network bandwidth is so cheap it's more cost-effective to create these virtual tunnels.
Does that help at all?
For instance, I have a VPN setup between my home network and my work network. At any time I can access home machines from the office and vice versa. This is very helpful for me as it allows me to wrap it up at 5:00 and then go home and finish any tasks that didn't get completed during the day. From home everything works just as it would from the office.