A Kippa or Kipa, also known as a Yarmulke, is a hemispherical or platter-shaped head cover, usually made of cloth, often worn by Orthodox Jewish men to fulfill the customary requirement that their head be covered at all times.
The tradition to wear a Kippah is not derived from any biblical passage. Rather, it is a custom which evolved as a sign of their recognition and respect that Hashem is always above us.
The Kippah in the Jewish law
The Talmud states,
Cover your head in order that the fear of heaven may be upon you (Talmud Bavli, Shabbat, 156b). Rabbi Hunah ben Joshua never walked 4 Amot (2 meters) with his head uncovered. He explained:
Because the Shechinah is always over my head (Talmud Bavli, Kiddushin, 31a). Jewish law dictates that a man is required to cover his head during prayer (Mishneh Torah, Ahavah, Hilkhot Tefilah 5:5). Originally, a head covering at other times for Orthodox males was a custom, but it has since taken on the force of law because it is an act of Kiddush Hashem (Shulchan Aruch, Orach Chayim 2:6). The 17th-century authority Rabbi David HaLevi Segal suggested that the reason was to distinguish Jews from their non-Jewish counterparts, especially while at prayer.
According to the Shulchan Aruch, Jewish men are strongly recommended to cover their heads, and doing so, should not walk more than four (4) Amot bareheaded (Shulchan Aruch, Orach Chaim, 2:6). Covering one's head, such as by wearing a Kippah, is described as honoring God (Shaar HaTzion, OC 2:6). The Mishnah Berurah modifies this ruling, adding that the Achronim established it as a requirement to wear a head covering even when traversing less than four cubits (Ber Heitev, OC 2:6, note 4, who quotes the Bach, Taz and the Magen Avraham), and even when one is standing still, indoors and outside (Mishnah Berurah 2:6, note 9, 10). The Kitzur Shulchan Aruch cites a story from the Talmud (Shabbat 156b) about Rav Nachman bar Yitzchok who might have become a thief had his mother not saved him from this fate by insisting that he cover his head, which instilled in him the fear of Hashem (KSA 3:6). In many communities, boys are encouraged to wear a kippah from a young age in order to ingrain the habit (Ber Heitev, OC 2:6, note 5).
More lenient opinions also exist, and many great rabbis did not wear a head covering. The GRA or Vilna Gaon says one can make a Berachah without a Kippah) and other Poskim, and wearing a kippah is only a Midot Chassidus.
Yarmulka Types and variation
In the Middle Ages in Europe, the distinctive Jewish headgear was the Jewish hat, a full hat with a brim and a central point or stalk. Originally used by choice among Jews to distinguish themselves, it was later made compulsory in some places by Christian governments as a discriminatory measure. In the early 19th century in the United States, rabbis often wore a scholar's cap (large saucer-shaped caps of cloth, like a beret) or a Chinese skullcap. Other Jews of this era wore black pillbox-shaped kippot.
Often the color and fabric of the kippah can be a sign of adherence to a specific religious movement. Knitted or crocheted kippot, known as kippot serugot, tend to be worn by Religious Zionists and the Modern Orthodox (source), who also wear suede or leather kippot. The hit Israeli TV series, Srugim, which has been compared to the US TV series Friends, takes its name from the knitted kippot worn by the main male characters.
Members of most Haredi groups usually wear black velvet or cloth kippot. In general, the larger the kippah, the more traditionalist the wearer. By contrast, the smaller the kippah, the more modern and liberal the person is.
More recently, kippot have been observed made in the colors of sports teams, especially football. In the United States, children's kippot with cartoon characters or themes such as Star Wars are popular. (In response to this trend, some Jewish schools have banned kippot with characters that do not conform to traditional Jewish values. (source)) Kippot have been inscribed on the inside as a souvenir for a celebration (bar/bat mitzvah or wedding). Kippot for women are being made and worn. A special baby kippah has two strings on each side to fasten it and is often used in a brit milah ceremony.