Zotero's sync feature can keep your library up to date on all computers that you use and can also double as a backup. 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 Zotero and be configured to sync to the server.
First, set up a (free, of course) 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.
To add a citation, click the first button ("Insert Citation") on the toolbar. Select the reference you want to cite and click OK. Zotero will add the citation at your cursor.
At the end of your paper, click the third button ("Insert Bibliography"). Your bibliography will appear, and new citations will be added automatically. Change bibliographic styles with the last button on the toolbar ("Set Doc Prefs").