Zotero works within Firefox, so you need to have Firefox 3 or higher installed before you can use Zotero.
Note that Zotero is continually being updated and comes in multiple versions; the 3.0 Beta version is still under development and might not work properly for everyone. However, it does have some significant features you might want to explore, such as compatibility with other browsers including Chrome and Safari.
For the current version, if you want to be able to work on multiple computers, you will need to install and run both Firefox and Zotero on a flashdrive. For more details, see the Zotero FAQ (via the tab above). It is also possible to sync your library and files so that you can access them from any computer on which Zotero has been installed.
For more information on using Zotero, watch the video below or read the Zotero Quick Start page.