It's a technology that allows you access Internet as if you were connecting from a different point in the world. This means, among other things that your local ISP/network operator cannot inspect or filter your traffic and that for any website or other service you will appear as having a different IP address.
Check out the (mostly) scientific diagram on the right we made to help you understand.
VPN stands for "Virtual Private Network". The term refers to a set of technologies that enable a computer to connect to a private network via Internet (or other public network) and exchange data with other computers on that network securely as if it was physically part of it. This is usually achieved by establishing an encrypted connection with a remote computer and "tunneling" all the traffic via that connection. While VPNs are used for many different purposes, "VPN service" most commonly refers to a provider of VPN connectivity that is used to tunnel all Internet traffic past the local ISP. That's exactly what ZenVPN offers.