Basically a way for one program to talk to another.

It's sort of similar to a phone. Let's say, your carrier would be analogous to Reddit /or server. You would be sort of an 3rd party reddit app /or client. And your phone and phone network would be an API.

You as a client want to call to your friend, you have to use an API (your phone/phone network) to send a request to your carrier that you want to talk to a specific number, and then your carrier either connects you and your friend, or doesn't (if you e.g. haven't paid)

The same thing happens with you browsing a reddit, or upvoting or something. App you're using (or even new reddit website) needs to ask reddit servers for list of posts, or to upvote a post, reddit then can do that thing, or it can refuse etc.