Pete, Tony,
Looks like I've let myself in for something here

NB I'm talking PC, Windows XP here, as that's what I have handy. The basics should be the same for Vista, or for older versions of Windows.
Your problem(s) probably relate to this bit:
You may need to select the correct sound source (eg Stereo Mixer, rather than Mic, Line In, CD In, etc) and adjust the recording level; this can usually be done via the control panel for your sound card.
where I deliberately skated over the details as they are going to vary from PC to PC depending on what sound card is installed.
You need to find the program that controls how your sound card (or onbound sound) is configured, and set options within that. Again, sadly, how you get to that program can vary too.
There may be an icon in the system tray which you can hover over, and then right-click on to run it. On the other hand, there may be an entry in the Start Menu which corresponds to the name or manufacturer of the sound hardware. Or an icon within the Windows Control Panel (again with name of the card or manufacturer). There might even be all three variations... all leading to the same place.
I've attached a screenshot of the control program for the onboard sound on the PC I'm typing this on, which shows the options relating to recording.
Master Volume: controls the overall volume and balance for all sound being played via Windows.
Recording Level: controls the recording level and balance while recording.
Recording Select: selects the source of the sound to be recorded. In this case I've selected Stereo Mixer, which is the sound playing through Windows. The other options here correspond to different input sockets on your PC (Microphone, Line In, CD drive, etc).
You need to play with the Master Volume, and Recording Level settings while the music is playing so that the bars showing the recording don't go into the red much (or at all). You can do all this while listening to what you want to record, but without Audacity running. Once things are adjusted, stop the streaming audio, start Audacity, hit the record button, and start the stream again. Once the stream has finished, hit the Stop button in Audacity, and save the music in whatever format you prefer. Once it's saved to disc, stay in Audacity and edit any extra from th start & end, and re-save.
You also need to make sure that a couple of settings within Audacity are correct:
Edit -> Preferences -> Audio I/O -> Recording
Device: Will probably be already set to Microsoft Sound Mapper - Input (this may well be OK). There may be another option in the dropdown list which refers to your sound card.
Edit -> Preferences -> Quality
Check what it says here. If you're going to burn to CD eventually, this should be set to 44100 Hz
Hope this helps...