Let's say you hurt your arm and go see the doctor. Both of these terms dictate what you owe, but they calculate it completely differently.
The Copay (The Flat Fee)
A copay is a hard, flat dollar amount. The plan says "$30 Specialist Copay". It does not matter if the doctor spends 5 minutes with you or 2 hours with you. You hand the receptionist $30, and you are done.
Coinsurance (The Percentage Split)
Coinsurance kicks in usually after you hit your deductible. It is a percentage split between you and the insurance company—usually 80/20. If your hospital bill is $1,000, the insurance pays 80% ($800) and you are billed for 20% ($200). Coinsurance is much scarier because you rarely know the total price until the bill arrives.