Q. Why do Jewish men cover their heads?
For men it is a sign of humility before God. The Talmud records that Rav Huna said he would never walk four cubits without a headcovering since “the Divine presence is above my head”. Rav Nachman bar Yitzchak’s mother said, “Cover your head so what the fear of God may be upon you.”
Covering the head became in time a mark of piety, though in the medieval period there were still some rabbis who recited the blessings and even the Shema bareheaded, though this was increasingly disapproved of.
Obviously the headcovering has to be visible. There is a question about whether some of the tiny kippot that are sometimes seen these days are adequate.
For women, headcovering is a sign of modesty before men. The sages speak of covering the hair as dat y’hudit, “the Jewish rule” (Ket. 7:6). There is strong disapproval of flaunting one’s body. Even showing off one’s hair can have a seductive message.
The various styles of headcovering for women include the sheitel or wig, though this is a relatively recent innovation. It must, however, be pointed out that sometimes a sheitel is so magnificent that it may appear to defeat the purpose.