I am primarily a backend engineer, so I have been at the receiving end of quite a few blank stares when I tell people I build APIs. The last time someone asked me, they had just added me on Snapchat, so I used the example of Snapchat to explain what I do.
The answer I gave (not the best one, but I tried my best to make a layman understand): The moment you tap the add friend button on my profile, there is a computer sitting somewhere is America that knows you have requested to be my friend, through a message sent from your phone to the computer in America. The computer should now do a bunch of operations and make sure that I as an end user know that you have added me as a friend and show me a notification. As an API engineer, I design and build those operations.