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