I could scarcely believe that this was available.. but it certainly is. You just have to get the settings right.
My 2012 I3 laptop has the Realtek chip to be found in, well, lots of laptops:
Having downloaded and installed Audacity, make sure your settings are like this, with particular attention to the Loopback option:
May 2017: I now have a laptop with a soundcard that has no StereoMix option. However, the recording still works without touching the soundcard options in Windows. This is how my options look on the new machine. It still works:
With that done, press the red recording button, then start the audio you want to record, and you are away:
Not directly relevant, but I’m finding right now that Audacity sits for an age (e.g. 2-3 minutes) on its splash logo before starting up properly.