Installing Zotero is a two-step process. On the Zotero download page, you will first need to download the standalone Zotero software for your operating system (Zotero is available for Windows, Mac, and Linux). Then, you will want to download the connector for your browser -- this will allow you to save content to Zotero directly from the browser with a single click.
Zotero offers plug-ins that allow you to automatically insert citations and create bibliographies from your Zotero library in Microsoft Word, LibreOffice, and Google Docs. The different word process plugins are available on Zotero's Word Processor Plugins page and under the Tools menu in Zotero once it is installed on your computer.
You can use the standalone Zotero client, browser connector, and word processing plugin without registering for an account on the Zotero website. If you want to synch your data between different machines or back up your Zotero database online, however, you will need to create an account on the website.