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 higher 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