I'm still not entirely sold on blending these two topics onto the same page. But they're both such niche topics (good for you for reading this, by the way) and I rarely think of one without the other, so for now they're together.
As I explain it to anyone that doesn't want to get too nerdy, an API is just the term used for two computer programs talk to each other. Like if you post a photo to Instagram and it shows up later on Facebook, it is because Instagram is using Facebook's API.
Like most things in computers, there are lots and lots of opinions on it. I do my best to avoid those holy wars, so you won't see me talking about them here.
Of the many places online that try to answer this, I'm most fond of Quora's answer. Their answer basically breaks down to:
Do you have other resources that I should know about? This is a place where I'm always craving new ideas & ways to think about the problem space. If you have suggestions, please share them.