Choose Zotero Standalone if:
Zotero Standalone (unlike Zotero for Firefox) runs as a separate program instead of part of your web browser.
It works with Firefox, Chrome and Safari.
You can install both Zotero Standalone and Zotero for Firefox and use both on the same computer.
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.)