Stop me and ask questions! (Although I may save them for the end)
Who and what is Xamarin and what do they do?
Mono - cross platform, open source .NET development framework
Mono allows .NET code to run on non-MS devices, such as...
Mac,Linux,BSD,iOS,Android,PlayStation 3 and 4and Vita,Wii
...and so on
If you are interested in making games with Mono, check out the excellent MonoGame.
Who and what is Xamarin and what do they do?
Xamarin Platform
Takes code written in C# (or F#) and runs it on a Mono runtime or compiles to native code
Xamarin.iOS
iPhone, iPad, and iPod Touch
Xamarin.Android
Android phones, tablets, watches, TVs, Google Glass, Kindle Fire phone and tablets
(and Blackberry 10 via Android compatibility, and Ouya and other Android game consoles, and that weird Nokia X line)
Xamarin.Mac
Macintosh computers by Apple, Inc.
Xamarin.Forms
Native UI for iOS, Android, Windows Phone with a single codebase
Test Cloud
Automated testing on hundreds of devices
Why should I care about the Xamarin platform?
Write in a common, modern language developers (may) already know
Reuse code on multiple platforms
Use Visual Studio (2010+) or the free Xamarin Studio
Many companies claim 60%, even 80% shared code. 95%+ possible with Xamarin.Forms
Commercial product with support - important for businesses (and governments)