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I'm curious about how this sort of animation is typically done code-wise.

A few examples:

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This is called inverse kinematics. Google is probably your best friend on this one as it can get complex.

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I get it, I have to calculate the motions necessary to achieve a desired position. In this case, I would only calculate the motions for the desired bone. –  Gerstmann Jul 25 '11 at 9:59
    
Indeed, the video description says "...an animation system based entirely on inverse kinematics..." –  Byte56 Jul 25 '11 at 22:43
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