A Rig is an animation control system for a Model. Rigs can use different methods for animating. For example it is possible to animate a mesh only by using empties and constraints, so no single bone is involved, yet it is an animation system.
The most common form of Rigs is based on Bones. This special form of a Rig is also named Armature.
Rigging is the process of creating the Rig.
However since in Blender we clearly have to distinguish the processes and workflows, we should also use precise terminology. Thus we distinguish between Rigging (make the rig) and Skinning (weight the mesh).