Basically, a kippah covers one’s head. While there is no mandate to do so in the Torah, one early reference to head covering is in Talmud, Tractate Kiddushin 31a, in which a rabbi would not walk bare-headed more than four cubits without a head covering, as the Divine Presence is above his head. The Shulkhan Arukh (Orah Hayim 2:6), makes this into a ruling.
Some say that the reason for covering is to show (and know) that God is over our heads (even more than “above”). Others say it is to show the distinction between God and human. Some say it is a sign of respect, others say it shows our fear of God. Some even say it shields us from the brilliant holiness of God, lest we take too much at a time.
This last is interesting, as it is reflected in Exodus 34:29-35. When Moses came down from Mt. Sinai he didn’t know his face was glowing (...ki karan or panav bedab’ro ito), and folks shied away, so he covered his face when not talking with God and when not speaking thereafter to the crowd. By the way, when Aquila Ponticus of Sinope translated the text into Greek (around the second century CE), he missed the word “or” (“light,” perhaps making a read-o). As the root of the word “karan” קרן)) is a cone shape, Aquila gave Moses horns emanating from his head rather than rays of light. That reportedly led to the notion that Jews have horns, which sadly persists to the modern day. But we digress.
Other peoples wear kippot: Muslims (the taqiyah, from the Persian for “dome,” or topi, Hindi for “cap”) and Buddhists, for instance, and Catholic clergy may be seen in them (zucchetti, meaning “half-pumpkin”), as well. Ancient Mesopotamians wore slightly more conical varieties to keep their heads cool after shaving them. The root of the word “kippah” means “dome.” The Yiddish word “yarmulke” is, some think, from the Aramaic “yira Malka,” “in awe of the King.”
My Uncle Mashe’s ability to slip his fedora off while slipping on the kippah was impressive.