Once you have created an account at Zotero.org, and installed the Zotero client on your device(s), you can link your account to each installation.
Once your installations are linked to your online account, your Zotero library will be automatically synced to be stored on the Zotero server. No matter where you work, your library will be saved to the Zotero cloud.
Not crazy about using Zotero's cloud storage?
If you are already using an online file storage service such as Dropbox, Box, or Google Drive, you can configure Zotero to store files and folders on those services.
In Zotero, go to Zotero Preferences -> Advanced -> Files and Folders. Here you will change your Zotero file storage Base Directory by using the Choose… option to go to the folder you just set up.