Zotero is Open Source software, and is available for free download at http://www.zotero.org
Be sure to install all the different components you'll need:
Zotero Standalone (the desktop application) for Mac, Windows, or Linux
The browser connector (add-in) for the browser of your choice
The word-processor plug-in for Microsoft Word or Open Office
Syncing your Zotero can ensure that you always have a backed-up copy of citations and will allow you to use and access your Zotero account across platforms or computers. Check out Zotero's Syncing guide for more information.
To join an existing group, you simply need to log into your Zotero account though the Zotero website. The group's creator can add you either by an email invitation or by using your Zotero username. Once you've logged in and confirmed, you're in the group.
Once you've added the group, you'll see it appear under the Group Libraries section of your Zotero window.
*If the group library doesn't appear, try re-syncing your extension or app. If that doesn't work, check the troubleshooting guide.
Zotero's group guide has more information about groups, how they work, and how you can create your own.
Check out these tutorials from other institutions!