Xamarin is primarily a platform for developing mobile applications (Android and iPhone) using C#.
The Mono project is an open-source port of the .NET framework. It is most commonly used to run .NET under Linux. From a performance standpoint, it runs about 50% (on average) as fast as the .NET framework.
MonoDevelop is an open-source C# IDE analagous to Eclipse or Visual Studio.
Miguel de Icaza was the primary developer for Mono and is one of the founders/owners of Xamarin.