What is a character rig?

Answer 1

It's a variety of virtual structures for controlling, in groups, the elements that constitute mathematical models, for ease and realism in animation.

Among the various structures available are joints (bones) which can be assembled in hierarchical structures, forming for example: shoulder --> elbow --> wrist --> knuckles; thus controlling the various vertices that compose the arm --> forearm --> palm --> phalanges respectively and through the appropriate "skinning" procedure.

This can be controlled in forward kinematic (FK) transmitting the changes from top to bottom of the list or in inverse kinematic (IK) calculating the rotation of the "bones" from a fixed position. For example, the rotation of the elbow would be calculated from the position of the hand going "inversely" to the elbow (if that's what you order your 3D program to do).

There are also lattices to perform deformations, usually applied for modeling too. Bezier lines for controlling rows of joints (bones) and empty objects to track movement to.