Faugère's F5 algorithm for computing Groebner bases. A huge number of algorithms in algebraic geometry rely on Groebner bases, and F5 is reputed to be "the best". But there is only one implementation, and it is not so widely available. The community could really benefit from a second, well-explained implementation.

This is more of a ``foundational'' answer that you might have wished for, I'm sure. But it seems that most algebraic geometry algorithms eventually rely on Groebner bases, so why not start there?