A counterfoil is the strip you keep when you tear or cut a cheque, ticket, or similar slip. The main part goes to the other person; the counterfoil stays with you as a record of what you paid, sent, or booked. You write or stamp the same numbers on both pieces so you can match them later. It is mainly a British word; American English more often says stub. You meet it in older banking and office talk, and on some tickets. Though actually, many people now pay by phone and never see a paper counterfoil, so the word feels a little old-fashioned — but the idea, “the bit you keep,” is still clear.