Download and install Zotero from the Zotero website.
NOTE: There's a different version for each operating system: Windows, Mac or Linux.
Connectors allow your web browser to save citations to your Zotero library.
The Zotero downloads page should automatically detect what browser you're using and give you a link.
NOTE: Each browser has its own version of the Zotero connector, so if you use Chrome, Firefox and Safari you'll need to install all three.
