Syncing your library means backing up your citations on Zotero's server. This way you can access your citation remotely and retrieve them if you are away from your computer.
This arrow will appear in your Zotero library. It will spin when it is uploading your citations.
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 Zotero and be configured to sync to the server.
For more details and help troubleshooting sync problems, check the Zotero site.