The difference between Android support V3 V7 V13

Simple translation:

V4 is used for 1.6 to a higher API.

V7 is 2.1 to a higher API usage, which contains actionbar.

V13 is 3.2 to a higher API for use.

Original:

V4 Support Library

this Library is a designed to being used with Android 1.6 (API level 4) and higher. It includes the largest set of APIs compared to the other libraries, including support for application components, user in Terface features, accessibility, data handling, network connectivity, and programming utilities.

v7 Libraries

there is several libraries designed to being used with Android 2.1 (API level 7) and higher. These libraries provide specific feature sets and can be included in your application independently from each Other.

V7 AppCompat Library

This library adds support for the Action Bar user interface design pattern.

Note:
This library depends on the v4 Support Library. If you are using Ant or Eclipse, make sure you include the v4 Support Library as part of this library‘s classpath.

So yes, need both jars if you want to use V7.

Read more about the support Library Features

Update for Android-support-v13.jar

v13 support Library

This library was designed to being used for Android 3.2 (API level) and higher. It adds support for the Fragment user interface pattern with the (Fragmentcompat) class and additional Fragment support CL Asses

When you see the package details it had a class fragmentcompat as given in definition. So it had not the classes of AppCompat library.