If the center of gravity shifts outside of the support base then the structure (or character) will fall. This is what happens during a walk cycle. Every time a leg is thrown forward to take a step we are falling a little and then stopping the fall when the leg lands on the floor.
For character designers, we need to be sure the weight is evenly distributed around the line of gravity. If the character pose is not symmetrical then the masses will be stacked in a way that transfers the line of support, balancing weight thoughout.