Go to http://www.zotero.org and click on the red Download button.
Download and install both the Zotero app and the Zotero connector.
Warning: Zotero does not work well with Safari, so it is better to use Google Chrome, which can be downloaded for free online.
Zotero will automatically identify your computer's operating system and browser and will give you the appropriate download links. If you wish to download other versions, click on the links under each respective download button.
When you first use the Zotero Connector in Chrome, it will show up underneath the Extensions button, which looks like a puzzle piece to the right of the address bar. You can then click the pin next to the Zotero Connector so that button shows up next to your address bar: