Rigging geometry with armatures mainly depends on the armature modifier, vertex groups and the armature.
You can manually add the armature modifier to a mesh object and select an armature. The mesh object will deform based on it's vertex groups. Each bone of the armature (with the Deform checkbox checked) needs a vertex group with the same name. It will deform the vertices in that vertice group based on the weights in the group.
When parenting a mesh object to an armature you have the option for vertex groups to be created automatically. When you rename the armature bones, the vertex groups will be renamed as well. When you rename vertex groups, armature bone names stay the same: the links will be broken. When you add new bones, vertex groups will not be created automatically and the bones will influence nothing. Meaning if you want to add the deforming bone bone_13.l you have to create the vertex group bone_13.l. Then you can select the bone in weight paint mode and the associated vertex group will be selected as well, ready to be weight painted.