An API (Application Programmable Interface) is a way that a developer can write client code that will interact with another piece of software. In the case of the web, most API's are ways to interact with the data that the software has available. In Facebook's API, you can interact with your profile (the data) in a couple of ways. You can send friend requests, like posts, and post statuses, but when you do these things, you do them in the language that the code understands. The Facebook website doesn't inherently understand what clicking a button is supposed to do. Facebook has programmed their website to talk to their servers in a represenation that their server can understand. Like ASCII is a way of sharing textual information via binary, a web API is a way of sending encoded information to a server. To start, lets look at the language they use.
When you make most web requests on the internet, an HTTP request is sent to a remote location, determined by IP address. The HTTP request has a predefined, agreed upon, and standard format. It usually involves a message header and a message body. The request involves a couple components, which I will go into more detail here.
First, it contains the URL that you are trying to reach. URL's are one of the most important parts of any web API. They give the remote server information on what you are trying to access, and how it should respond.
Next, you have an HTTP verb. There are many HTTP verbs, but the only ones that are standard in web API's are GET and POST. A GET request does just that, it gets information in many different forms. When you view a website in a web browser like Chrome, Firefox or Safari, the browser makes a GET request for all the URL that you gave it. Then, it proceeds to GET all the information it needs to display the page. POST is the other side of the coin. It is used to send information to the remote server. In the web browser, forms where you input data usually make a POST request to the server with the information that you put into the forms within. These both have the same usage in web API's.
As an example, take a look at the Twitter API documentation. 
As you can see, the API contains both a VERB and what is called an endpoint.
Endpoints are another form of a URL that omits the base URL. For this API, the
base URL is https://api.twitter.com/1.1/, so a URL would be formed by
appending the endpoint to the base URL. The end result would look like
https://api.twitter.com/1.1/statuses/retweet/:id.json. In the Twitter API
specifically, the :id is just a placeholder for the information that you
are giving to the server. In this case, the server wants an id of the tweet to
retweet.
URL's have a couple parts to them. This diagram shows the structure, taken from
tutsplus.com
