Both terms refer to your share of the costs of a covered healthcare service.
"Co-insurance" is a portion of the allowed amount of the service. For example, if your plan's allowed amount for an office visit is $100 and you've met your deductible, then your coinsurance payment of 20% would be $20.
"Co-payment" is a flat fee. For example, your plan may have a $50 co-payment for you to see a specialist.