No, Web applications run in the server.
Desktop Applications run in the client.
This only works because you're running it locally, but it will not work when deploying the ASP.Net Web Application to a Web Server.
If you need to create Web Applications, then use the standard ASP.Net techniques which deliver HTML [+ JavaScript + CSS] content to the browser. You may choose between "traditional" ASP.Net WebForms, or ASP.Net MVC.
If you need to create Windows Desktop Applications, use WPF.
If you need to create Windows 8 "Metro Style" Applications, use WinRT XAML.
winforms is completely useless at this point.