When Zotero Connector is installed in Firefox, you should have a Zotero icon in the top right of the browser that looks like a Z, a piece of paper, or possibly a book (it depends on what you're looking at in your browser).
By default, Chrome seems to group browser extension beneath a dedicated Extensions icon in the browser toolbar. You can make sure there is a dedicated Zotero icon in your browser toolbar by clicking the Extensions icon, then clicking the pin next to Zotero:
If you can't run the desktop version of Zotero, you can follow these steps to get the Zotero Connector to sync with the web version of Zotero.
Now when you click the Zotero Connector icon, whatever you're looking at will be saved to your Zotero web library.