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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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up vote 4 down vote accepted

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 " animation system based entirely on inverse kinematics..." – Byte56 Jul 25 '11 at 22:43

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