The popular story says handicap comes from disabled beggars holding a cap in their hand. It is a tidy explanation, and the evidence says it is wrong.
Wordorigins.org calls it "an inventive, but incorrect, stab at the word's origin." Snopes calls the etymology false. It notes that "cap in hand" is a separate phrase, recorded from 1565 as a sign of respect to a judge, with a begging sense first clearly recorded in 1887.
What the record shows is a wagering game. Etymonline dates handicap to the 1650s, "from hand in cap." Two bettors agree a wager on an unequal contest and call a neutral umpire to set the odds. Each puts a hand holding forfeit money into a cap.
The umpire announces the odds, and the bettors withdraw their hands. Full hands mean the odds were accepted and the bet stands. Empty hands mean the bettor declines and forfeits.
The name first appears in print in 1653, in George Daniel's Idyllia. The practice is older. Wordorigins.org traces an early form to Langland's Piers Plowman around 1390, where an umpire rules on a trade of a cloak for a hood and adds a cup of beer to the deal. The page says the method took the name "new fair" from a London market.
Samuel Pepys wrote on 19 September 1660 of a "Handycapp" game he had never known before. Wordorigins.org says it is unclear whether he meant this game or a card game.
Horse racing came next. Wordorigins.org cites a "Handy-Cap Match" in the Sporting Kalendar of 1751. Etymonline gives 1754 and says the umpire ruled that the superior horse should carry extra weight, which was called a handicap.
Wordorigins.org finds the figurative sense in the British Medical Journal on 7 September 1872, which called a discrepancy "a serious handicap in the race for priority." By October 1888 the American Annals of the Deaf used it for a person's disability.
Etymonline dates "handicapped" to 1915, originally used especially of children. So the beggar story gets the order backwards. The word was a wager long before it described anyone's body.