A mat is a generic term for a piece of fabric or flat material, generally placed on a floor or other flat surface, and serving a range of purposes including:

providing a regular or flat surface, such as a mouse mat
protecting that which is beneath the mat, such as a place mat or the matting used in framing and preservation of documents and paintings
protecting that which is above the mat, such as a wrestling or gymnastics mat, or an anti-vibration mat
changing the state of that which passes above it, such as a doormat attracting dirt from shoes

In domestic settings:

A doormat is a device normally placed in close proximity to a building's entrance - immediately outside or inside the threshold - and is designed to enable people to wipe the dirty soles their shoes before entering the building. Doormats are often made of Coir, a coconut fibre; palmyra (palm tree) fibres and stalks, as well as nylon, rubber, cloth, aluminium and other materials. Doormats are also known as welcome mats in part because they are situated at the front foor at which visitors would be welcomed; also because it has long been a tradition to print messages - such as Welcome - onto doormats. This in turn has given rise to a subculture of cartoons featuring characters returning home to find (to the reader) a humorous message on the doormat.

A bath mat is a device used on the floor of a bathroom to provide a warm non-slip surface, and to absorb small amounts of water, much like a towel.

Place mats or serving mats are flat piece of fabric or other material used on a table at the points at which dishes and plates will be loacted during a meal. One common purpose of such mats is to provide a thermal and phsyical barrier between that which is places on the table, and the table itself - for instance, to stop hot objects from changing the physical appearance of the table.

More generally:

a mouse mat is a small flat mat used to provide a flat surface on which a computer mouse can be operated. (A mouse employing a ball which is driven by friction and which drives a x and y movement sensors requires a regular flat surface with high enough friction to drive the ball. Older optical mice - typically those from Sun Microsystems - required a mouse mat with a differentially reflecting grid.)

an anti-vibration mat is a mat which performs the function of isolating vibration between that which is above the mat and that which is below. Such a mat might typically be used when mounting a heavy machine which vibrates, onto a concrete floor; in the absence of the anti-vibration mat, the machine tend to erode the floor through abrasion. Contrawise, some machines, such as the lithography machines used in chip fabs, need to be isolated from vibrations in their locale, and so are mounted on anti-vibration mats.

a slipmat is a circular piece of slippery cloth or synthetic material designed to allow disc jockeys to turn or stop vinyl records on record players, or to scratch.

Chat-Mat is a recordable talking doormat that speaks messages when stepped on. Chat-Mat was invented in 1995 by Ralph Baer, the inventor of the first commercial video game system.

In psychology:

Doormatism - Treatment of a human as such that he or she lets another step on passively without any objections. See Casper Milquetoast.