This is going to a bit confusing to do, though it is possible. What exactly do you need to record from the internet? Anyway, I would suggest getting the Applian Freecorder. It's a handy little toolbar that sticks onto your internet browser. Whenever you need to record sound (eg, you're listening to a radio article that's interesting and wanna hear it again), you type in a name for the file, and hit the record button. Whatever you're listening to will be written onto a .wav or MP3 file for you to listen to later. Hope that helps.
Right, a lot of the new integrated sound cards don't offer that option, maybe for cost-cutting reasons or maybe for DRM reasons.
One customer got around this by running a cable from the line out on his sound card to the line in. Heh heh.
The integrated sound cards that come with your computer are often not very good sound cards. If you are serious about recording, spend some money on a good solid sound card.