To use Zotero, you'll need to install two things:
Both of these are available from the Zotero download page.
Download and install Zotero from the Zotero website. (There's a different version for each operating system: Windows, Mac or Linux.)
Below is a screenshot of the Zotero application.
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.
Below is a screenshot of the Zotero Connector icon that will appear on browser bar after installation.
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.