Cloth armor, referred to as "padded armor" in Dungeons and Dragons and other fantasy games, is without a doubt the most common form of armor in the history of mankind. More people have gone into combat wearing fabric armor than any other type, and in most wars the majority of combatants have gone into battle wearing fabric armor.
There are a variety reasons for this popularity. Cloth is cheap, light, and very flexible. It doesn't restrict the wearer very much at all, especially compared to other forms of armor. It is also easy to produce and transport.
There are two primary approaches to providing protection using cloth. First, one can quilt the cloth with heavy padding in order to cushion crushing blows. Quilted garments were worn by themselves as armor as late as the 17th century, as evidenced by the long quilted coats used by some American colonists in place of the more expensive leather buff coat. Quilted garments were often used underneath mail and plate armor, to cushion blows and ease chafing.
The second approach is to use tough fabrics to provide protection against penetration or cutting. This approach is used even today, in modern kevlar armor. Heavy canvas or the equivalent was often used for this purpose.
One of the most extreme forms of cloth armor used historically was the ancient Greek linothorax. This armor consisted of up to thirty layers of linen or cotton pressed together and stiffened with glue. This produced a rigid, vest-like garment barely more flexible than a metal breastplate.