To use Zotero, you'll need to install two things:
Both of these are available from the Zotero download page.
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.
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.