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.

Note: The Term “Rigging” is often also used as a synonym for Skinning (see below). We also find the term “Rigged Mesh” for example, which would actually be more accurately named “Skinned Mesh”.  It is also very common to talk about “Rigged Characters”. Depending on the context it is often clear what is meant.
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).