Max flow and Matching problem
Based on the book "Algorithm Design" by Kleinberg and Tardos, Kevin Wayne has a set of slides (second part here) where the following applications of max flow and matching are presented in some detail: 


survey design
airline scheduling
image segmentation
project selection
baseball elimination
assignment problem
input-queued switching


For even more examples, have a look into the exercises of Chapter 7 of Kleinberg and Tardos' book, or into the exercises of the book "Network Flows: Theory, Algorithms, and Applications" by Ahuja, Magnanti and Orlin.