Much harder and more time-consuming than sprite ripping, sound ripping involves recording the sound or music that you want, and editing it for whatever purposes you need it for.
Firstly, you'll need a copy of the ROM you want to rip a sound from, and an emulator with a sound logging function. You'll also need a program to edit the sound with, but if you have Windows you should have a copy of Sound Recorder on there somewhere, which works well enough.
If you don't know any emulators with sound logging functions, here's a list of the ones I know:
So, load up the ROM you want to rip the sound from. You'll have to find the sound logging option (sometimes called audio logging) yourself, and when you select it you'll have to specify where and under what name you want the sound to be recorded.
It is advised that you have a bit of 'dead weight' on the file (sound before and after the bit you're recording) to make it easier to edit. Ripping a sound effect is easy enough but it has to be the only sound playing or you'll record a segment of the background music as well. Conversely, when you're ripping music make sure no sound effects play or they'll be recorded too.
Music ripping is a lot more difficult; first you have to listen to the track several times to determine where it loops. Once you've worked it out, start recording a couple of seconds before the loop, and record one single loop before stopping a couple of seconds after the loop begins again. This results in a single loop of the sound track with a little bit of dead weight on the ends.
In the folder you specified there should be a sound file with the filename you gave. Open up sound recorder and try to find the very start of the loop by playing the sound. It may take a few tries and a lot of moving the position bar around, but eventually you should be able to find the start of the loop. Select "Delete before current position" and the dead weight at the start will be gone.
Play through the rest of the file and repeat the process to determine where the loop ends. Once you've found it, pick "Delete after current position" to remove the dead weight at the end. Now save it to a different directory to keep the original intact in case you made a mistake.
Here's the big problem: You need a program that loops sound files seamlessly, so windows media player is out of the question. What files do do that, though?
Oddly enough, I use RPG Maker 2000, as it loads the start of the sound file right when it comes close to the end of the file, looping without any hiccup. Of course, there are bound to be better programs out there, so if you find one feel free to use it.
If you've gotten the loop right the file should be able to play continuously without any obvious loop. If this is the case, you can delete the original... though you now have a major problem if you want to distribute your sound file.
Sound files are logged in WAV format - a seamless sound format that creates absolutely enormous files. The best option is to convert the sound into an MP3 file with a suitable sampling rate - test them out to find the best one. 48kbps works for most of the files I've recorded from GBA games.
Mind you, I don't actually HAVE an MP3 converter. I use Flash, because it automatically converts WAV files into MP3 files when a Flash movie is published. This allows for the addition of preloaders and start/stop buttons, but I'd just use MP3s if I had any choice.