The Twilio Cloud Module is a JavaScript library that makes it easy to use Twilio for mobile app development, allowing developers to integrate Twilio features into their Parse apps with ease. The tutorial demonstrates how to send SMS using the Twilio Cloud Module and Parse Cloud Code function, making it simple to add secure communication functionality to mobile apps without needing servers. The example app uses a native iOS application written in Objective-C, which presents a table view of friends and allows users to make calls to their friends via SMS, demonstrating the flexibility of integrating Twilio with Parse for mobile app development.