A knowledge base article about What's the difference between a public and 1918 IP address for my bIT virtual server? provided by the UC Berkeley IT Service Hub - Knowledge Portal
In general, devices assigned a public IP address can reach resources off campus and have the ability to accept incoming requests from anywhere on the Internet.
Devices assigned a private (aka 1918) IP address cannot get to resources off campus and can only accept incoming requests from the Berkeley campus network including the campus VPN and wireless networks.
On a related note you may see hostnames like [VPS name].1918.berkeley.edu that include 1918 in the name. These are legacy entries. As of 2017 campus DNS will no longer include 1918 for hostnames and aliases.