A yarmulke (pl: yarmulkes) is a special head covering worn by Jewish men as a sign of religious piety, The Hebrew word is “kippak” (pl: “kippot”), the common English equivalent being skullcap, as it usually covers only the top of the skull. A yarmulke (pl: yarmulkes or yarmulkas ) is simply the name for a yarmulke in Yiddish, and the word is often used by English speaking Jews. The two terms will be used interchangeably.
The Torah does not demand that all Jewish men cover their head. The earliest written references to the religious requirement of covering the head are to be found in the Babylonian Talmud, dating from 500 CE/AD. The process of the custom’s acceptance in all Jewish communities has taken over fifteen centuries, and its application is still a moot point today. Whereas in the not-to-distant past most Jewish men observed the custom during all their hours of wakefulness, nowadays some observe the custom at all times, others only during religious ceremonies and yet others only when partaking of food. Furthermore, while most Jewish men do not wear a yarmulke at all, some Jewish women in some congregations do wear a yarmulke, at least during religious ceremonies at the synagogue.
What, then, does a yarmulke, or a yarmulke, look like? In other words: what is it made of, how big is it, what colors may it have?

Yarmulke materials

There is no limitation as to the material of which a yarmulke is made, and any material is befitting to serve as a head–cover, whether generally or when reciting blessings or prayers. The Talmud specifies that an appropriate head cover may be one made of wool (Tractate Hullin 138 A) or of any other cloth, whether woven, knitted or crocheted. A yarmulke may also be of cane fibers (Tractate Shabbat 120 A) or straw. If the head cover is a straw hat, it is appropriate even if the total area between the fibers is greater than that covered by the straw (Rashi’s comment on Isaiah 7, 3; Responsa Hattam Sofer part 6, B). Journalist Dov Ganhovsky has taken this idea of areas between the fibers to the extreme and suggested a yarmulke with exceptionally large loops that is actually of joined holes (Yediot Aharonot daily newspaper, January 28th 1988). Germany, and northern France (Ashkenaz).
As of the 10th century, Jews in Europe could cover their head with a soft barret, and from the 13th–14th century on, a yarmulke would be made of thin linen (Rabbi Yehudah ben Asher, Responsa Zikhron Yehudah, 20). Later, when velvet became cheaper, yarmulkes were also made of velvet. During the 20th century, people began to wear a yarmulke made out of new materials. Some turned to yarmulkes crocheted out of DMC thread, while others wore a yarmulke made out of upper leather. Lately it has become fashionable, especially in the United States, to wear a yarmulke made of suede, what in Israel is called an “American yarmulke”.
The rule that a yarmulke may be of any material explains why some synagogues have at their entrance, for the benefit of passersby without a yarmulke who may enter, a stock of yarmulkes made of Bristol paper, thin stiff cardboard or thick paper, or why people, in moments of religious emergency, cover their heads with a (clean!) handkerchief. Placing a hand on one’s own head is not an appropriate head cover, as head and hand are parts of the same body and the body cannot cover itself (Shulhan Arukh 91, 4; Rabbi Moshe Isserlis (=Rama) on Shulhan Arukh 74, 4). However, pulling out one’s sleeve to cover the hand and then covering the head with hand and sleeve is a head cover. Likewise, placing one’s hand on another person’s head is an appropriate head cover for the other.

yarmulke use in ancient times

Color of yarmulkes

There is no clear directive as to the color of a yarmulke, and it may be of any color or combination of colors. A person wearing a plain black yarmulke is generally recognized as presenting himself as being in a state of religious impurity or of mourning, whereas a white one symbolizes festivity and religious purity. A white and blue one combines the Jewish national colors. A yarmulke may also be of any other color or hue, this showing personal preference. Some people have adopted the fashion of wearing a yarmulke having exactly the same color as the hair so it wouldn’t be obvious. Journalist Dov Ganhovsky has taken the idea of color to the extreme and suggested a transparent yarmulke (Yediot Aharonot daily newspaper, January 28th 1988).
A yarmulke may carry a design that can be of the same color as the yarmulke or in a completely different, contrasting color or colors. The design can be words, such as in Name yarmulkes, a slogan, a company’s logo, in Hebrew or any other language. The design may also be a symbolic picture, whether concrete or abstract. It may be an integral part of the yarmulke’s texture, one that is knitted or crocheted into it, one that is embroidered on it or painted or printed over the yarmulke as in Printed kippot.

Size of yarmulkes

There is a considerable difference of opinion on whether there is a minimal size for a yarmulke, below which it cannot serve as an appropriate head covering. Among those who expressed the view that the size of a yarmulke was of little significance, were Rabbi Hayyim Yosef David Azulai (=Hiyda, 18th century, Eretz Yisrael and Italy) as well as today’s Rabbi Ovadiah Yosef (20th –21st century, Israel ). Others have demanded that the whole head, or at least most of it, should be covered by the yarmulke. Among them are Rabbi Shalom Mashash (19th–20th century, Morocco and Israel) and Nissim Cohen (20th –21st century, Israel). And then there are those who hold an in-between opinion on the appropriate size of a yarmulke, whether in principle or as a practical compromise. Among those are Rabbi Shelomoh Kluger (=Maharshak, 18th -19th century, Ukraine) and Rabbi Moshe Feinstein (20th century, Russia and USA). A fourth group are those who complain about the prevailing bad situation of yarmulkes that are too small, something like “being with but seen to be without”. To be mentioned are Rabbi Ovadiah Hadayah (20th century, Israel) and Rabbi Hillel Posek (20th century, Israel). There are, then, various customary sizes for a yarmulke, ranging from those that cover the top of the head almost completely to those that are barely visible.

Updated Nov. 3, 2013

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In the following video a talk about different head coverings in Judaism by Rabbi Jonathan Ginsburg:

Read the definition of Kippah in Wikipedia