Zotero runs as a separate program on your computer.
You can install Zotero Connector so it works with Firefox, Chrome and Safari.
Download and install Zotero just as you would any other program. You will also need to install the Connectors to allow your browser to save items to your Zotero library.
Connectors allow your web browser to save citations to your Zotero library.
Go to the Zotero download page and install the connector for your browser. It will automatically select the connector to install based on the browser you're using.
Each browser has its own version of the Zotero Connector, so if you use multiple browsers you'll need to install connectors for each one.