An API or an Application Programmer Interface is the collection of public, or exposed, methods/functions available for a programmer to use when interacting with an application, framework, or library. That is, an API always belongs to something; depending on what it belongs to, we can generally anticipate that it has a specific type of structure.
The primary way the term API is used is to describe a set of endpoints that an application exposes for the purpose of allowing a client to perform certain tasks regarding the resources associated with that application. Those tasks are often known as CRUD tasks:
The word API is also used when talking about the public methods that a library or framework exposes to the programmer. For example, you might hear someone say, "The new version of the jQuery API exposes some new methods." It is less common, though still quite frequently used.
APIs are the backbone of modern web development, so it's no wonder that standards and conventions have been designed for them. These standards and conventions primarily address the first type of API, describing the endpoints of an application.
A very common convention is REST. Read more about that here.