Capture the currently playing audio

Audacity can do it, but not with all sound cards. Hopefully yours will work with it, give it a try.

I use dbPowerAMP converter for sound capturing, CD ripping and format converting (it supports virtually almost all known formats) and Audacity for editing.

Both are excellent free programs. And between the two of them, you don't need much else for amateur sound recording and editing.

Note that the dbPowerAMP mp3 output is not free (comes with 30 days trial) due to mp3 licensing issues but the application is so good that it is well worth the 14$ for the license (and it comes with a power pack of few minor extra features).

If you want to record PodCast I would recommend Audacity.

Otherwise you can use an excellent program called HarddiskOgg. This is taken from the HarddiskOgg Website.

"HarddiskOgg takes a wave input stream from any Windows 95/98/2000/XP compatible sampling device (including microphone input and line in) and converts it to an Ogg Vorbis/Wave/Monkey's Audio/MP3 (optional) stream. This happens in realtime, so basically it is a harddisk recorder in Ogg Vorbis."

PolderBitS

Consider capturing the stream instead of the sound output.

this can be done using hinet recorder, for example - for asf/wmv streams. I'm sure there are other open source, freeware, solutions out there. 

Notice that some other online services aren't actually streaming audio, but downloading the entire mp3 file to your hard-drive (such as pandora.com, for example), for these stations you can (and should) use a specialized capturing utility - such as Pandora's Jar.

Good luck.