Xamarin.Android application lifecycle is the same as normal Android app. When talking about lifecycle we need to talk about: Application lifecycle, Activity lifecycle and Fragment lifecycle.
In the below I'll try to provide a good description and way of using them. I obtained this documentation from the official Android and Xamarin documentation and many helpful web resources provided in remarks section below.
Interesting links to broad your knowledge about Android application lifecycle:
https://developer.android.com/reference/android/app/Activity.html
http://www.vogella.com/tutorials/AndroidLifeCycle/article.html
https://github.com/xxv/android-lifecycle
https://developer.android.com/guide/components/fragments.html
https://developer.android.com/guide/components/activities/activity-lifecycle.html