You can sync your library to Zotero.org in order to have all your sources available on multiple computers. To do this:
Zotero will sync your library automatically. You can disable automatic syncing, and instead sync manually by choosing the button on the right end of the Zotero toolbar:
Zotero.org offers a limited amount of storage space for free. There are many options for syncing only item data, without sync full text such as PFs, or for syncing full text content using other services or Zotero plugins. For full information, see sync information from Zotero support.
Don't rely on syncing to back up your Zotero library! Be sure to back up the files in your Zotero library just as you back up other important files.
To find where your Zotero data is stored, open Zotero's preferences, choose the "Advanced" tab, then choose the "Files and Folders" sub-tab. The location will be shown under "Data Directory Location." Make sure this location is part of your regular computer backups!