Each bone in an armature deforms only a part of the mesh - so each vertex of the mesh has to know how much each bone influences it. They are just vertex groups with the names of the bones, but are called bone weights because of what they represent:
They are created automatically when binding (Set Parent CtrlP) the mesh to the armature with automatic bone weights:
The vertex groups will be generated only for bones that have Deform enabled in their properties. If you later add bones to the armature you can re-generate the groups again with binding the mesh again with the same parenting action.
But usually you can get better deformation if you edit the weights manually. That's done in Weight Paint mode. You can view the bone weights and paint them with brushes - and thus control precisely what and how each bone deforms:
You can select them from the vertex groups list or better select some bone in pose mode, then shift-select the mesh and go into Weight Paint mode. You will see the weight group for that bone. You can also move, rotate and scale the bone with G,R and S keys to see how it deforms while you are painting. You can also select a different bone without leaving Weight Paint mode with ctrl-selecting another bone.
There are also tools to specifically help with weight-painting under the Weight tools in the Tool Shelf.
The Weight Gradient tool is very good for making linear weigh transitions - to linearly blend the influence between two bones, something that will make nice weights for the twisting if you wouldn't be happy with the automatic weights.