The most interesting aspect of Game Theory I have encountered are the "Evolutionary Stable Strategies". Not sure if they are even officially considered Game Theory, as I read about them in a Dawkins book.

But I think the concept is extremely important when trying to understand the world around us. It makes the "people are just not rational" aspect irrelevant - evolution in the long run is rational, or rather mindlessly testing all the relevant strategies, so we still need Game Theory to figure out which strategies will be the most likely to prevail.