You need to have each bone "assigned" to the mesh with weight painting. The easiest way to do is is to have Blender do it for you.

In object mode, select the mesh, and then the armature. (the order matters)  Then, hit <kbd>Ctrl</kbd>+<kbd>P</kbd> to parent the armature to the mesh.  Use the option, "with automatic weights".  Please note that will clear all previous weight assignments.

I hope that works for you!