A virtual private network is a method for connecting your computer securely to the network of an organization on the other side of the Internet. When you use a VPN, all of your computer's Internet communications is packaged together, encrypted and then relayed to this other organization, where it is decrypted, unpacked, and then sent on to its destination. To the organization's network, or any other computer on the wider Internet, it looks like your computer's request is coming from inside the organization, not from your location.
VPNs are used by businesses to provide secure access to internal resources (like file servers or printers). They are also used by individuals to bypass local censorship, or defeat local surveillance.