I thought Activity was deprecated
No.
So for API Level 22 (with a minimum support for API Level 15 or 16), what exactly should I use both to host the components, and for the components themselves? Are there uses for all of these, or should I be using one or two almost exclusively?
Activity is the baseline. Every activity inherits from Activity, directly or indirectly.
FragmentActivity is for use with the backport of fragments found in the support-v4 and support-v13 libraries. The native implementation of fragments was added in API Level 11, which is lower than your proposed minSdkVersion values. The only reason why you would need to consider FragmentActivity specifically is if you want to use nested fragments (a fragment holding another fragment), as that was not supported in native fragments until API Level 17.
AppCompatActivity is from the appcompat-v7 library. Principally, this offers a backport of the action bar. Since the native action bar was added in API Level 11, you do not need AppCompatActivity for that. However, current versions of appcompat-v7 also add a limited backport of the Material Design aesthetic, in terms of the action bar and various widgets. There are pros and cons of using appcompat-v7, well beyond the scope of this specific Stack Overflow answer.
ActionBarActivity is the old name of the base activity from appcompat-v7. For various reasons, they wanted to change the name. Unless some third-party library you are using insists upon an ActionBarActivity, you should prefer AppCompatActivity over ActionBarActivity.
So, given your minSdkVersion in the 15-16 range:
- If you want the backported Material Design look, use
AppCompatActivity - If not, but you want nested fragments, use
FragmentActivity - If not, use
Activity
Just adding from comment as note: AppCompatActivity extends FragmentActivity, so anyone who needs to use features of FragmentActivity can use AppCompatActivity.