Application Programming Interface is a term used to describe the functionality exposed by an application that allows it to be used through a program. For example, you can use Inventor's API to write a program that will perform the same types of operations you can perform when using Inventor interactively.
Having an API is important because it allows you to add functionality to Inventor that is specific to your needs. Inventor, by necessity, is a general CAD system, meaning that it's not aimed at any specific industry or used to model only certain types of products. By providing an API, Inventor allows you to add additional functionality and optimize repetitive operations to make it more productive for your individual needs.
By providing an API, Inventor also provides you the ability to better integrate Inventor into your overall Enterprise process. For example, you might write a program that interfaces with your company's inventory database to obtain the current price for components so that the price shown in a part list is always up-to-date. You might also write a program that extracts data from assemblies to provide MRP systems with their required information. All of this can be done manually, but by automating it using a program you're able to significantly increase productivity and minimize errors.
An API is also important for the reason that it allows third-party applications to integrate with Inventor. It's through the API that products for PDM, NC, and FEM are able to interact with Inventor.
