This is like choosing your favorite friend, I love a lot of algorithm's in their own, special way. I could also answer in families of algorithms, but that would be cheating. Some of my favorites have already been mentioned, too.

So, I'll mention Monte Carlo integration. It's very simple to implement, it was one of the first tasks my first CS professor gave freshman students, and he did it for the same reason I love it; it gives such profound insights about how computers can solve complex mathematical problems humans can't. I shiver every time I solve a problem with a Monte Carlo method.