I gave a Lunch and Learn about Xamarin at the Logical Advantage office on September 20th, 2016. I contributed to "everything you need to know about Xamarin", a Q&A article about my event, which was posted to the Logical Advantage blog leading up to the event.
Xamarin is a framework to develop a cross platform mobile application using C#. Developers have many choices for building mobile applications. There are many frameworks and platforms to pick from like Ionic, Swift, Objective-C, and Java for Android. Xamarin was really strong, but now its gaining more momentum since Microsoft has purchased them ($400M) and is offering as part of Visual Studio and MSDN subscriptions. You may wonder what the benefit of Xamarin is compared to other cross-platform tools. Most cross platform tools use web technologies which do not look like native apps and incur significant performance costs compared to coding in the platform's native language. The C# code that you write in Xamarin will perform as good or sometimes better than code that you would write in the native languages.