A killer app is an program so useful that people will buy a larger system just to use it. For example, the VisiCalc spreadsheet was a killer app for the Apple II, maybe the first program to be called a “killer app.” People would buy an Apple II just so they could run VisiCalc. Microsoft Office is a killer app for Windows: many people run Windows so they can run MS Office.
I was thinking about the mathematical analog of killer apps. For example, finding maxima is a killer app for calculus. If that’s all you could do with calculus, we’d still teach calculus. After one semester of calculus, students can easily solve optimization problems that would be virtually impossible otherwise.
Mechanical vibrations are a killer app for differential equations. (Non-mechanical systems such as LRC circuits follow the same equations.) If an engineer applies anything from a differential equation class, this is probably it.
Contour integration is a killer app for complex analysis. There are other applications of complex analysis, but contour integration certainly one of the big ones.
Lack of killer apps
Some areas of math don’t have a killer app that I can think of. They may have practical application, but there isn’t a particular application that stands out, not one that many people would agree on . If you did a Family Feud-style poll on applications of calculus, differential equations, and complex analysis, I image the examples above would be on the board if not the top result.
Category theory can be useful, but its applications are scattered. If category theory has a killer application, I doubt there’s a consensus of what that application would be.
A killer application is different from a key theorem. I imagine a lot of people would say that the Yoneda lemma is the most important theorem in an introductory course in category theory, but I wouldn’t call it a killer app. My idea of a killer app is something that fills in the blank “You should take a course in X if for no other reason than that you’ll be able to ______.” For instance, many people take a course in statistics just so they can do linear regression.
If you have ideas about what killer apps would be in various areas of math, please share them in the comments below.
 I’m reminded of someone’s description of G. K. Chesterton as a master who left no masterpiece. That is, he wrote a lot of great lines, but no great book.