Zotero's Groups feature allows you to share references with other Zotero users online. It's a great way to work on collaborative research projects.
Syncing your library means backing up your citations on Zoter's server. This way you can access your citation remotly and retrieve them if something happens to your personal 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.