You can move your Zotero library between computers or programs using the export function.
On Computer A:
On Computer B:
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.
Install Firefox and Zotero on a USB drive and you can take your library with you to use on any computer.