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On-demand web-site compilation mode means that when you edit page and then hit F5 (run with debugging) or Ctrl-F5 (run without debugging) the solution will compile all of the class library projects, then compile the /app_code directory and Global.asax file, and then instead of re-verifying all pages within the web-site it will only verify the current page you are working on, and any user controls that the page references. Note that ASP.NET will automatically re-compile any other page or control you access at runtime - so you will always have an up-to-date and current running application (you don't need to worry about old code running).
To enable this, right-click on your web-site project and pull up the project properties page. Click the "Build" tab on the left to pull up its build settings. Within the "Build" tab settings page change the F5 Start Action from "Build Web Site" to the "Build Page" option: