API stands for Application Programming Interface.Actually it connects two application by exchanging information between them.
Example:If you want to install an application on your site to retrive Domain Authority(DA) of domains then, you have to request an API from MOZ because they are the actual provider of these service.By using API user website will send a request to MOZ that you want to know DA of some domains.Then MOZ will send you the DA of those sites and your sites visitor will be able to retrive the DA from your site.