In this Numberphile video Dr. Emily Riehl introduces stable marriage problem.
Practical applications of Gale-Shapley algorithm are much less romantic. Variations of the algorithm is used to, for example, optimize assignment of users to servers in content delivery networks. Similarly optimal matchings are often need in economic and management sciences. In fact, L. Shapley was awarded Nobel Memorial Prize in Economics in 2012 (together with A. Roth) for the theory of stable allocations. One of the most well known examples of this is matching graduating medical students to their first hospital appointments.