What is the difference between Theme.AppCompat and Theme.Material in Android?

This is the raw material design theme, which is used by Android 5 and up_{Not sure how this works wrt. new Android libraries for app design}:

<style name="AppTheme" parent="android:Theme.*">

This is a way to use material design on pre-lollipop devices, which maintains compatibility.

<style name="AppTheme" parent="Theme.AppCompat.*">

You can design for newer APIs using AppCompat and still have it work on earlier API levels than what the base level for material design is.

In this case, it essentially means that you can run material design on platforms that predate material design. This isn't as important anymore now that versions that early make up at most ~2.5% of the market share at the time of writing.

Note, however, that using AppCompat does give you additional compatibility helpers beyond just being able to use the material theme on older devices. Also note that AppCompat has since been deprecated in favor of whatever system is currently mainstream and that Google hasn't axed yet (Jetpack?), which may or may not work differently.

<style name="AppTheme" parent="Theme.AppCompat.Light.DarkActionBar">    
<style name="AppTheme" parent="android:Theme.Material.Light.DarkActionBar">

Theme.AppCompat is for above API 7+

Theme.Material is for above API 21+