A rig (also armature, the terms will be used interchangeable) is the skeleton structure used to control and animate meshes.
Each armature is a hierarchical connection of bones (similar to a meatspace skeleton — the foot bone's connected to the leg bone).
Control
Via rotation and translation of individual bones, the armature controls the pose of the connected mesh. You can read more about this on the Meshes and Armatures: Rigging page.
Deform
Via scale, the armature deforms the connected mesh. This is how custom rigs are used to change V's body shape.