The problem is that Skype wants the recording input to be set to Microphone, but to record sounds playing on the computer Audacity needs the recording input to be set to Stereo Mix. This conflict makes it difficult or impossible to do both at the same time.
The recommended solution for recording Skype is to use dedicated Skype recording software.
Google turned up "MP3 Skype Recorder v.3.1" as the first hit when I searched for
free skype recorder.
I've not tested that software myself so try at your own risk, but it could be helpful for others if you could write back and let us know if it works satisfactorily. Here's the link:
http://voipcallrecording.com/
There is also "
Pamela" which is not free, but is widely reported as working well (free 30 day trial version available).