This post is to explain how to setup and use NuGet in Xamarin Studio and use NuGet packages in your Xamarin.Android project. I'll also explain how to create NuGet packages for Xamarin.Android too.
NuGet, since version 2.5, supports Xamarin.Android and Xamarin.iOS as its "target framework", as well as other targets such as .NET 4.x, Silverlight, Windows Phone and so on.
The same should apply to Xamarin.iOS, so just replace "Android" part with "iOS".