An API is a set of instructions that programs can use to talk to each other. In web development, these instructions are usually defined as a set of urls that can take different parameters.
To make a Twitter bot, you might use an API to collect boring data and do something interesting to it. For example, @how2butt uses the url http://www.wikihow.com/api.php?action=query&list=random&rnnamespace=0&rnlimit=1&format=json from the MediaWiki API to find a random WikiHow article. Then, in order to make the world a better place, it replaces one of the words in the article's title with "butt."
Let's use a simpler example to look at how an API works. Try visiting the url http://pokeapi.co/api/v2/pokemon/1/ in your browser.
This url is an endpoint of the Pokéapi. It has extensive documentation at https://pokeapi.co/ which describes all the different kinds of urls you can use to see Pokemon data.
The pile of text you're seeing is JSON, a data format that most programming languages can understand.
"I'm machine-readable!"
We can use Python's requests and json libraries to read from JSON urls like this one:
Now all you have to do is pass the text you've created to Tweepy. You can read data from pretty much any API just using requests.get() and json()!