wit is reasoning power — the mind as the seat of understanding and sense — and, in the sense that now dominates, the ability to relate seemingly disparate things so as to illuminate or amuse, a talent for banter or persiflage.
The older sense is the original one. Old English wit or witt comes from Proto-Germanic and ultimately from the Indo-European root weid-, "to see," extended metaphorically to "to know" — the same root that produced wise, vision, and idea. For most of the word's life it meant mental capacity outright: understanding, intellect, consciousness, sound mind. That sense survives almost entirely in fixed expressions and in the plural, which never made the shift the singular did: at one's wit's end, keep your wits about you, scared out of one's wits, live by one's wits. Each of these is about functioning faculties, not about being funny.
The comic sense appears in the 1540s, denoting the ability to connect ideas and express the connection amusingly, and from there the quickness of intellect in speech or writing that the word now chiefly names. "A wit," meaning a person of learning or verbal brilliance, is recorded from the late fifteenth century. Wit is narrower than humor: it is verbal, compressed, and cerebral, turning on precision and surprise rather than on warmth, and it can be cutting in a way humor generally is not. It is also a faculty rather than a form, which distinguishes it from 📝satire — satire is a genre with a corrective aim, and wit is one of the instruments it uses.
The verb wit, "to know," from Old English witan, is now archaic. It survives almost solely in the phrase to wit, "namely" — attested by the 1570s as a loan translation of an Anglo-French phrase used to render Latin videlicet.
"Wit without employment is a disease." ~ Robert Burton, Anatomy of Melancholy, 1621