The Xamarin Studio IDE will disappear now that Visual Studio for Mac (which is a fork, and kinda the evolution of Xamarin Studio) has been released.
Mono is an implementation of the .NET Runtime on platforms not yet officially supported by Microsoft (.NET Core is currently beeing developed by Microsoft for that purpose). Basically, this allows you to run intermediate language (IL, in a DLL) compiled from C#, VB, F#, on those new platforms.
Xamarin is a company (now owned by Microsoft) that produces Xamarin.iOS/Xamarin.Android which are a set of .NET bindings to the (originally in Swift/Obj-C or Java) native APIs. This allows you to call all the UI, Sensors, low level stuff on iOS and Android from .NET Code (C#, ...).
Monogame, for mobile platforms, is a framework (.dlls) based on mono for running its logic, but also on Xamarin for initialization (creating the base UI, interracting with sensors, ...) stuff on iOS/Android. That's why a Monogame mobile project is a Xamarin project.