Sometimes you want the privlidges you get when you're on campus computers, but you don't want to be on campus. Maybe you're working from home, your favorite coffee shop, or on the road. VPNs allow your computer to appear to have the IP address of another network. This means is you get the benefits that you normally get on campus, from anywhere. That's right: full access to libraries, journals, and even live streaming sports from sites like ESPNu.
In addition, operating under a VPN can offer additional security when using un-protected WiFi, and using VPN can get you around firewalls that are commonly used to block sites like Facebook at some corporate offices (Or, in foreign countries). This is also useful for using remote desktop to access your computer from a different network.
Most schools offer a couple options for VPN clients. To find what your school offers, do a google search for your school's name and then "VPN client." Usually the first one or two links are from your school's IT office, and will give you instructions on how to use VPN. My preferred method is to use Cisco's VPN client (I know Clemson and Colorado, as well as many other schools offer this), although many people prefer the in-browser VPN, which is also offered by almost all higher education institutions.
To use VPN, you must log into the network with your username and password, so this tool is only good for current university students and faculty. I think there are a couple of free VPN clients for non-students, but I can't vouch for these.
The downside to VPN is it will slow down your internet connection while you are using it.