Quicksort ( especially implemented in a functional language like haskell ). It's a perfect mix of an elegant algorithm and a beautiful implementation. Simple, elegant, beautiful and powerful. What more can you want in an algorithm.
Came into the comments to find this, and I'm going to expand on your description: I feel like quicksort isn't [uniquely] beautiful except when used alongside bubble sort to explain computational complexity (both big-O "worst case" and big-Ω "best case"). These were the examples that really made the concepts "click" for me.
