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.
Open Chrome or Safari and click the Zotero Connector link for that browser to install it.
Each browser has its own version of the Zotero connector, so if you use both Chrome and Safari you'll need to install both.
(Firefox does not require a connector to work with Zotero; it's built into Zotero for Firefox.)