API is short for Application Programming Interface. Basically, a software company will create an API that allows developers to "safely" read and write data from the underlying database of the application. It is safe in the sense that the software company makes sure you can't mess up the business logic or allow access to confidential information that you are not supposed to be able to access.
A desktop API allows a programmer to write applications that programatically access funtions of a desktop software applications. Custom Apps 360, LLC uses Intuit's SDK (Software Developers Kit (which is another name for an API)) in order to read and write data to QuickBooks through code.
A REST API allows a programmer to write applications that programatically access funtions of an application through the web. Custom Apps 360, LLC use REST APIs to send text messages using Twilio or Trumpia. We use REST APIs to run Mozenda Screen Scrapers to grab data off of websites. We use InfusionSoft's REST API to send customer information up to their Online Customer Relationship Management System.
A WebAPI allows Custom Apps 360, LLC to create website like apps that provide functionality to other applications that call it. It is like the reverse of an API or REST API. Instead of the code "Consuming" services from other applications, our WebAPI's provide services to other applications. For Example: When we programmatically send a text message using Twilio, the person receiving the text message may reply to it on their smartphone. Twilio wil send the reply to our WebAPI which will automatically perform whatever service we instruct it to.