Quilts are multi-layered textiles, usually created using three different species of fiber: a woven or knit cloth outer layer, a layer of batting or wadding and finally a woven or knit back. The process of combining these layers together into a single textile is what we call quilting (hence the name of the finished product).
Quilting dates back a very long time; the earliest recorded evidence comes from a carved depiction of what is likely a quilted garment on an ivory figurine from Egypt dated to 3400 BC. We also have a recovered Mongolian quilted floor mat that dates back to between 100 and 200 AD.
In Europe, quilting was not a common practice until roughly the 12th century, when it became seen as a vital trade for supporting the crusades through the production of gambesons, aketons and arming doublets - rugged textiles that were used to complement maille armor. When the crusades ended, these textiles evolved into fashion items (in particular, the doublet became a staple of upper class male clothing all the way from the 14th through the 17th century.
During the colonization of America, a method of quilting known as paper piecing became quite popular among settlers, used to more easily replicate desired patterns onto a quilt:
Today, most paper pieced projects have the paper itself torn out and discarded at the end; in colonial times, however, the paper was often left incorporated into the finished piece. Given the rarity of paper products at the time, settlers often used the backs of old envelopes, letters and pamphlets for creating their patterns - a practice that has led to old quilts becoming a primary source of information about early pioneer life.
Hand quilting has waned with the advent of machine sewing, but is a practice enjoying a small renaissance throughout the world as a means of relaxation and pragmatic hobbying. Quilts are great long term projects that will create something that will likely be used for whole generations in your family; if you ever thought about getting one started, there's no time like the present.