The Vonage SMS API is used to send text messages programmatically with Vapor framework. A Vonage API account is required for this tutorial, and the necessary dependencies are installed in Xcode 12 using Swift 5 or greater. The project is set up by creating a new Vapor project and adding a .env file to store API credentials. A web page is created to enter phone numbers and messages for SMS sending, and model structs are defined to conform to the Codable protocol. The /send route is defined in the routes function to parse form data, make an HTTP call to the Vonage SMS API, and return the response status to the view renderer.