To teach is to cause someone to know something — to impart knowledge or skill by instruction, demonstration, or example.
The verb takes two objects comfortably, which most English verbs do not: one teaches a person, and one teaches a subject, and both can appear at once. Used without an object it names an occupation rather than an act. A third sense turns punitive — "I'll teach you to make fun of me" — where what is imparted is the disagreeable consequence of an action rather than any skill, and an obsolete sense meant simply to show someone the way.
That older sense is the original one. Teach descends from Old English tǣcan, "to show, point out, declare," from a Proto-Indo-European root meaning "to show" that also yields token and Latin dīcere, "to say." The root image is a finger indicating something, not a vessel being filled — teaching as directed attention, with the pupil still doing the looking.
The pairing with 📝learn is not symmetrical, and the asymmetry is the useful part. Teach names an attempt; learn names an outcome. A lesson can be taught to a room in which nothing is learned, and a great deal is learned that nobody taught. Early Modern English blurred this by letting learn also mean "teach," a usage now nonstandard.
"To teach is to learn twice." ~ Joubert
"When one teaches, two learn." ~ Robert A. Heinlein