<a href="http://en.wikipedia.org/wiki/Pythagorean_triple#Geometry_of_Euclid.27s_formula">Wikipedia</a> explains how to come up with the parametrization (I think I've seen the term "generating functions" for this, but really that term means something entirely different).
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<b>Update:</b> Ha! apparently you can also use Hilbert's theorem 90 to obtain the parametrization (Elkies' <a href="http://www.math.harvard.edu/~elkies/Misc/hilbert.pdf">note</a>).