There are two options: backing up and syncing to Zotero Cloud. It is a good idea to back up regularly. Zotero will automatically sync to the cloud if this option is checked in Zotero preferences.
To Back up go to File on the Zotero menu bar and select Export Library.
This will open a dialogue box. You want to save the file in RIS (a bibliographic citation file developed by Research Information Systems). If you have added Notes on Zotero, make sure the Notes box is checked. If you want to save your full-text files, check the Files box too and click OK. This will ensure both your notes and files are exported.
This will save your library of references as a 'My Library.ris' file. It is useful to have this backup in case you are working offline and/or if you cannot connect to Zotero Cloud. To download the file, have to Import the file into an empty Zotero Library.
If you're regularly using more than one computer in your research, Zotero's sync feature can keep your library up to date on all of them. Zotero can store a copy of your library on the Zotero.org server and check it for updates whenever you open your library on a different computer. All your computers must be running the same version of Zotero.
First, set up a (free, of course) Zotero.org user account. Then:
Repeat this configuration on each of your computers. Any updates you make on one of your computers will be reflected on the others. This even works to synchronize your library among Windows, Mac and Linux computers.
For more details and help troubleshooting sync problems, check the Zotero site.
Set up automatic syncing in Zotero preferences so that any changes to your Zotero Library syncs to the Cloud. Back up your Zotero library by exporting it as a file and save this on a different computer or to the network drive at UCL.