Xamarin is a Microsoft-owned software company that is responsible for allowing the creation of cross-platform implementaions of the Common Language Infrastructure using the C# language.
Basically, Xamarin allows a developer on the Windows platform using C# and Visual Studio to build apps that work/run on Android, iOS and Windows.
Xamarin came around when developers wanted to know if the new .NET Framework (June 2000) would or could ever run on Linux. A lot has happened since then and .NET is now fully cross-platform and open-source. Xamarin, once an independent company, was acquired by Microsoft and the Xamarin platform was added to the Visual Studio available packages as an free optional plugin. I use Xamarin within Visual Studio to allow me to create cross-platform apps. So when I create an app for Android I create a mirrored project for iOS and Windows, all sharing the same code.
If you are interested in learning more about Xamarin please follow this link to their Wiki entry.