1. A portion of a certificate, ticket, label, advertisement, or the like, set off from the main body by dotted lines or the like to emphasize its separability, entitling the holder to something, as a gift or discount, or for use as an order blank, a contest entry form, etc.
2. A separate certificate, ticket, etc. for the same purpose as above
3. One of a number of small detachable certificates calling for periodic interest payments on a bearer bond
4. Metallurgy: A sample or metalwork submitted to a customer or testing agency for approval
Coupon as "certificate of interest due on a bond" is first attested in 1822 from French coupon, which is literally "piece cut off" from couper ("to cut"). The original coupons were a set of certificates that were attached to a bond paper. The bonds ran for a term of years, so over time the certificates were to be cut off and used to get interest payments.
Our modern sense of coupon, as in those slips of paper you cut out of newspaper inserts, is first attested in 1906. Coupon-clipper pre-dates that and originally referred to someone who held a lot of coupon bonds.
Now we have this disaster on TLC: