A co-pay is a kind of financial responsibility. Most health insurances and even supplemental and government sponsored health plans obligate the member or patient to some kind or some degree of financial responsibility. Co-pay requirements will vary depending on the type of benefit that the health insurance is currently called to cover— for example, many health insurance HMO products do not charge a patient a co-pay when the insurance subscriber utilizes a kind of preventative care benefit— such as seeing the doctor for an annual physical.
Want to know why? See: "Why health insurances don't usually charge co-pays for preventative care."
What is a co-pay?
What is a share-of-cost?