Download and install Zotero from the Zotero website.
Only Windows, Mac, and Linux are officially supported by Zotero at this time. Linux users should check to see if Zotero can be installed using their distro's package manager before downloading it from the website. Android and iOS users should check the Zotero Knowledge Base at https://www.zotero.org/support/kb for information on third-party apps known to work with Zotero.
Connectors allow your web browser to save citations to your Zotero library.
The Zotero downloads page should automatically detect what browser you're using and give you a link.
Each browser has its own version of the Zotero connector, so you will need to install a separate connector for each browser you intend to use with Zotero. Only Chrome, Firefox, Safari, and Edge are officially supported by Zotero. Other browsers may work to varying degrees but are not officially supported. If you don't use one of the supported browsers, search the Zotero Knowledge Base at https://www.zotero.org/support/kb or the support website for your browser to see if there's a workaround or third party plugin you can use.
To use Zotero, you'll need to install two things: the Zotero application itself and a connector to allow your browser to save citations to Zotero. Both of these are available from the Zotero download page.
If you don't want to install anything or don't have permissions to do so, you can use the web-based tool ZoteroBib. It's similar to other web-based tools like Citation Machine and EasyBib.