MonoDroid allows developers to create native Android applications using C# and the .NET framework. It works by using managed callable wrappers to consume native Android APIs, allowing apps to access device features like contacts, GPS, and cameras. MonoDroid can be installed on Windows or Mac and includes the Android SDK, Xamarin Studio IDE, and libraries needed to build Android apps. The IDE supports common Android objects like activities, intents, views, and bundles to handle tasks like passing data between screens and storing app configurations.