Zotero comes in two forms-- a downloaded version and an online version sometimes referred to as Web-based Zotero.
Downloaded Zotero has more power and features. When you register for a Zotero account and sync through Zotero, you'll be able to use Zotero on multiple computers and your library becomes stored/backed up and available online through Zotero on the Web.
Zotero on the Web is a way to use on Chromebooks, public computers, mobile devices and tablets, or installing your library on a new computer. Register for an account and login to begin using it. Step by Step Guide from Carleton