Download Audacity, and, if you want to save your downloads in MP3 format, the LAME MP3 encoder.
Install Audacity and the encoder. Open Audacity.
Click on the drop-down menu on Audacity's mixer toolbar. Select "Wave Out" and "Stereo Mix" or "What U Hear" (The exact name may vary depending on your computer, sound card, and operating system). This will make Audacity take your sound card's output as its input.
Go to the streaming-radio website of your choice. Begin play and click on Audacity's "Record" button.
Click Audacity's "Stop" button when you are finished. Select "File" and then "Export as WAV" or "Export as MP3," depending on your preference. Save your recorded audio.