Stock insurers are corporations owned by shareholders of the firm. The shareholders hire managers to run the company and the insurance product is sold to customers who may or may not be shareholders in the firm.
Mutual insurers are companies owned by their customers.
Any policyowner of the company also owns a portion of the company. Reciprocal insurers or reciprocal exchanges are insurance companies in which policyowners agree to insure one another. They are very similar to mutual companies. HMOs essentially provide prepaid health care coverage. Once you pay your premium you can use the services of the HMO at little or no further cost to you.