The final component of a mattress is the fabric that is used to encase everything else and this too can play an important role in your mattress.
This fabric, called the ticking, can contribute to the performance, feel and durability of your mattress in several ways besides simply aesthetic appeal.
One of its main functions of course is to help maintain the shape of the mattress and to help protect the materials in the mattress from exposure to light, ozone, solvents, or other influences that can oxidize or degrade them more quickly.
In some cases a fabric can be a weak link in a mattress and wear out before other layers of a mattress. This however is rarely the case with higher quality fabrics. They can also play a role in the breathability of the outer layers of your mattress and aid in temperature regulation. In terms of breathability and moisture and temperature regulation, fabrics made with natural fibers tend to outperform fabrics made with synthetic fibers such as polyester. Some fabrics have special additives to help control temperature and moisture (Outlast and Coolmax being 2 of these) for those who tend to sleep hot. Fabrics also tend to vary widely in their "elasticity" with knitted fabrics being much more elastic than their woven counterparts ... especially those that are tightly woven. Flexible and durable fabrics are an important part of allowing the underlying layers to form a cradle and relieve pressure so it is important that the ticking of your mattress is both durable and flexible. How it is made and how tightly it is attached will also play an important role in altering the qualities of your mattress. Fabric tightness, like quilting, can also be used as a form of zoning however this too is better done in other layers of a mattress.
Fabrics can also be a part of having a more "natural" mattress as there are many available which are "organic" and very high quality. Semi synthetic fabrics like the many types of viscose/rayon materials and natural or organic cotton are popular choices here. Fabrics can also be replaced if they wear out before the layers of a mattress as there are many sources for zippered mattress replacement covers, with and without quilting layers, that can be used to recover your mattress. This can also be done professionally at a reasonable cost as well so all is not lost if you have high quality layers inside your mattress and the cover wears out before they do.
Finally, the "hand" of a fabric (how it feels and drapes when you handle it or move your hand over it) and its appearance are the final 2 properties of a mattress ticking and while we all want a mattress that "looks really nice" when it is not covered by sheets, these qualities play little role in the performance of your mattress even though they are often promoted as justifying the higher costs of some mattresses.
To recap all of this, the best mattress tickings are made of a high quality woven or knit materials that are flexible, durable, comfortable, and breathable and will allow the underlying layers do the job they were designed to do. Good looking is a bonus.