Note that it is a TWO STEP process to download Zotero. You must first download the app (1), and then download the browser extension for the browser you want to use (2).
Screenshot of Zotero.org/download, indicating the two steps that must be taken to fully download Zotero.
You will be prompted as part of this process to go to Zotero.org and register for a free account. You are not required to register for an account, but I recommend it.
Once you have created your username and password, you can go to the 'edit' menu in your Zotero app (NOTE- may appear in the 'Zotero' menu for Mac users), go into the Preferences menu, select the 'Sync' menu, and ensure that your username and password are filled in so that your library will automatically back up to the server, and sync across multiple computers.