Zotero changed a bit this year with the update to Zotero 5.0. This video will walk you through downloading the app and the new browser connector.
The default is for Windows and Chrome but there is an app for Mac and Linux as well as a connector for Firefox and Safari. You can find the Zotero download page here.
The video above outlines how to add the Zotero plugin for Word but there are a number of plugin options for everything from LibreOffice to LATEX to Drupal that will help connect to any tool that you use. Check out the entire plugin list here.
To ensure that all your accounts are synced together so that you can work from your browser to the web platform to the app, you need to make sure your accounts are synced with your account information. The video above will walk you through the steps for syncing your accounts.