Zotero's sync feature can keep your library up to date on multiple computers. 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 and be configured to sync to the server.
To synchronize data from different sources, you must first set up a Zotero.org user account.
Open Zotero Preferences (via the gear menu) and select the Sync tab. Enter your Zotero user name and password. Check the "sync automatically" box. Zotero will upload your library to the server.
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, but does not back up attached files.
To back up your attached files, you may choose to store your Zotero library on Zotero File Storage or a WebDAV enabled server such as webfiles.washington.edu.
For more details, alternative syncing solutions and help with troubleshooting sync problems, check the Zotero syncing help page.
Without enabling the syncing feature, your Zotero library is stored in your Firefox profile on your hard drive. Once you create an account with Zotero, you can synchronize what is on your server and your online Zotero library by clicking on the sync button on the top right hand size of the Zotero three-column panel. (You can also set automatic syncing as a preference.)
By logging into your account at the Zotero website, you can access your library at any computer.
Without logging into the Zotero website, you can also retrieve your Zotero library by installing Firefox and Zotero on a USB drive that you can use at any computer.
First: download and install Portable Firefox for Windows or Mac on your USB drive. This is a special edition of Firefox designed to run entirely from a portable drive. It keeps its own settings, bookmarks and Zotero library separate from the settings installed on your computer.
Second: copy your Zotero library from your computer to your USB drive.
One way to do this is to open Zotero on your computer, click the "Actions" (gear) button and choose Export Library (as in the image below). Save your library to a file anywhere on your computer. Then close Firefox, run Portable Firefox from your USB drive, open Zotero, click the "Actions" (gear) button and choose Import. Import the file you just saved.