ASP.NET applications are web application, they deliver HTML thru HTTP from IIS. It runs as a program called by IIS when a web request is received for it. You call the application thru a web browser.
A WinForms application is a windows desktop application that has a windows UI. It runs locally. You call it from the windows desktop.
A Console application doesn't have a UI per se, it runs in a command prompt window.
A class project (or class assembly) doesn't run anywhere. You have to write another application (WebForms/WinForms/console) that consume the classes that live in the class assembly.
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Peter