There is no simple answer to your problem. However, for starters, you might look into how audio fingerprinting works. This paper is an excellent start written by the creators of shazam:
http://www.ee.columbia.edu/~dpwe/papers/Wang03-shazam.pdf
I'm not sure how well that approach would work for dog barking, but there are some concepts there that might prove useful.
Another thing to look into is how the FFT works. Here's a tutorial with code that I wrote for pitch tracking, which is one way to use the FFT. You are looking more at how the tone and pitch interact with the formant structure of a given dog. So parameters you'll want to derive might include fundamental pitch (which, alone, might be enough to distinguish whining from other kinds of barks), and ratio of fundamental pitch to higher harmonics, which would help identify how agressive the bark is (I'm guessing a bit here):
http://blog.bjornroche.com/2012/07/frequency-detection-using-fft-aka-pitch.html
Finally, you might want to do some research into basic speech recognition and speech processing, as there will be some overlap. Wikipedia will probably be enough to get you started.
EDIT: oh, also, once you've identified some parameters to use for comparison, you'll need a way to compare your multiple parameters to your database of sounds with multiple parameters. I don't think the techniques in the shazam article will work. One thing you could try is Logistic Regression. There are other options, but this is probably the simplest.
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