A beret is a soft, round, flat-crowned hat

Hat

A hat is a head covering. It can be worn for protection against the elements, for ceremonial or religious reasons, for safety, or as a fashion accessory. In the past, hats were an indicator of social status...

, designated a "cap", usually of woven, hand-knitted wool

Wool

Wool is the textile fiber obtained from sheep and certain other animals, including cashmere from goats, mohair from goats, qiviut from muskoxen, vicuña, alpaca, camel from animals in the camel family, and angora from rabbits....

, crocheted cotton, or wool felt

Felt

Felt is a non-woven cloth that is produced by matting, condensing and pressing woollen fibres. While some types of felt are very soft, some are tough enough to form construction materials. Felt can be of any colour, and made into any shape or size....

, or acrylic fiber

Acrylic fiber

Acrylic fibers are synthetic fibers made from a polymer with an average molecular weight of ~100,000, about 1900 monomer units. To be called acrylic in the U.S, the polymer must contain at least 85% acrylonitrile monomer. Typical comonomers are vinyl acetate or methyl acrylate...

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The modern beret is derived from the Greek pilos, which came to be known in Rome as the pileolus and later, beretto. It was first mass-produced in 19th century France and Spain; countries with which it remains associated. Berets are worn as part of the uniform of many military and police units worldwide, as well as by other organizations.
Beret (originally berret and berretto) is from the diminutive form biretum of the Latinized Greek word birrus, which was the name of a knit sailor's cap.