Click on the armature that you are using, then without deselecting go down to the mode drop-down, and change that to pose mode.
Then select the object that you want to move, and in the mode drop-down, select Weight Paint Mode. Just paint the entire thing red, and it should work correctly. Completely blue means it won't move at all, red means it moves completely with the armature.
This ball I have will become a jumbled mess when I move the armature, because it is only half shaded. If I paint the entire thing red, it will move with the armature correctly, without stretching.