If you are on a Mac I would suggest using Genymotion (free for individual, paid for commercial use). If you are on Windows I recommend you use the Visual Studio Android Emulator. You must download the iOS Simulator for Windows if you want to see the iOS Simulator on your local Windows machine, otherwise you can remote desktop into your Mac to see the iOS simulator. However you can develop completely develop on Windows, you must connect to a Mac when you want to build your iOS solution or visually design in iOS. Visual Studio is recommended when developing on a Windows and Xamarin Studio for developing on the Mac. You can use Visual Studio or Xamarin Studio. The tooling required to create Xamarin applications is also a little more complex due to the ability to build for numerous platforms and licensing agreements that keep iOS building only available on a Mac. This is for developers who know how to code but are just starting their Xamarin journey.
Xamarin also supports game developing and graphic intensive applications via UrhoSharp (3D/2D) and SkiaSharp (2D) but I won’t cover them here.
Here I just focus on the common app building of Xamarin. Its an eco-system not a product and Xamarin’s/Microsoft’s offering is actually incredibly deep and wide. I have wanted to create this post for a while as I have never found an all in one place for a high level overview of everything Xamarin has to offer.