Thanks for the clarifications in the comments above. As I understand it, you are creating a map by logging into ArcGIS Online, choosing My Content, then Create Map. From here you are adding a layer using Add , and trying to configure the layer's popup to show a website's contents, rather than just a hyperlink to the website's address.
I don't believe this is possible using the built-in ArcGIS Online map viewer. The viewer allows you to configure a few options for a popup:
...but these don't seem to allow HTML. In order to display the contents of the website, you'll need to include it within an iframe, and this doesn't appear to be supported by the popup - in my testing, the HTML code is rendered as text, instead of being executed: