The Viterbi Algorithm (http://en.wikipedia.org/wiki/Viterbi_algorithm) 
is my all-time favorite, because of what it does: help machines 
understand the cause of a sequence of observations. Its real-world 
applications are humbling.

A strong contender is Alpha-Beta pruning 
(http://en.wikipedia.org/wiki/Alpha-beta_pruning), which makes computers 
better than humans at chess.

Otherwise, anything on 
http://www.scriptol.com/programming/list-algorithms.php is worth looking 
at.

-- follow-up on Viterbi real-world use cases --

When I was a student I implemented two systems that used the Vitterbi 
algorithm as school projects:

* A voice codec for the GSM. They use convolutional codes for channel 
coding. The Viterbi algorithm is used to decode them.

* A basic speech-to-text system that used the Vitterbi algorithm to 
solve Hidden Markov Models.