Tableau Server runs as its own set of processes, it can run on your own server or in a cloud. Users then access the server via web browser, iPad, Android device, or Tableau Desktop to browse visualizations created by Tableau Desktop.
Tableau Desktop creates visualizations and publishes visualizations to Tableau Server. Tableau Desktop can also export packaged workbooks which can be opened by the (free) Tableau Reader application that runs on Windows, similar to how Adobe Reader opens PDFs. The user experience of using Tableau Reader has the same sort of interactivity as what is delivered by Tableau Server, only in a native app and lacking the sharing features offered by Tableau Server.