A certificate of authority is a legal authorization from a state government to conduct business within the state. Certificates of authority are commonly needed for insurance companies or health maintenance organizations. In the context of the law, it can be illegal for companies to operate in states that they don't have certificates of authority from (if such certificates are required).
The reason why certificates of authority are often required is because the state government wants to make sure that the business that will be operating in its borders is properly qualified. The state government also wants to make sure that the business is able to follow all of the laws related to its industry. This process can be especially important in the health care field, where people's lives will be on the line and the quality of service is highly important.