Zotero requires that you install two different applications on each computer you use for research:
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.
If you have any problems, check the Zotero installation page.
Plugins for Microsoft Word and the Libre / OpenOffice are included in the standalone install. These allow you to easily cite items from your Zotero library in your papers.
Be sure to return to the Zotero download page to install the appropriate connector for your browser.
The page will detect the browser you are currently using but you can install connectors for Firefox, Chrome, or Safari.
Browser connectors will allow you to add items to your Zotero library automatically while you are viewing them online.