When I parent my gun mesh with Armature, in the vertex group I only see root vertex group. There is no other ones.
If I add them manually, they don't move my geometry.
I'm using my own mesh and did it exactly like you show on video. When I add groups manually and assign to parts of my mesh, they don't move it. Only root bone works.
Oh I see! It's because the armature modifier takes one armature as an input while all of your bones are separate armatures. Quick fix is to select all your armatures, make the root the active object, and press Ctrl+J to join.
For more about working with armatures and bones, I would definitely recommend Wayne's new rigging course:
https://cgcookie.com/course/fundamentals-of-rigging-in-blender-course
Hope that helps!