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 of Zotero.
First, set up a (free, of course) Zotero.org user account. Then:
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.
For more details and help troubleshooting sync problems, check the Zotero site.
If you're working on a public computer, there's another easy option for taking your Zotero library with you.
In your local Zotero library, click the Zotero gear menu, choose Export Library, and save the file to your USB drive. On the public computer, click the gear menu, choose Import, and find your Zotero file to import it.
When you're done, make sure to export any changes and import the new file onto your computer. Many public computers will erase your data when you log off, but use caution and delete your data from any public computer you're unsure of.