Bobby gave a good description of Authority, but only one meaning of 'mandate'. Perhaps it is used differently in North America? 'Mandate' is more often used here to mean an implied authority, rather than an explicitly stated authority. A government will often be described as having a mandate to enact rules, if it stated that it would enact those rules as part of its election manifesto. As a government it obviously has the authority, but it also has a mandate if it was elected based on those promises. It could also be used to describe the scope of enquiry of an investigating body. "The commission considered that investigating child abuse in non-state run institutions was also part of its mandate".

Authority is a status given by someone or something to a person or thing that gives them power to tell others what to do. For example, at work I may have the authority to give refunds to customers. The authority I have to do this comes from my manager. Someone who is new to our company may not have the authority to give refunds yet. Another example: In the United States Congress has the authority to create laws. Within our country, the citizens give congress the authority, or ability, to create laws. Now the word mandate is something that is required of someone. For example, at work we are mandated to take an hour lunch. The government may mandate that we pay our taxes. A mandate is a sort of command or enforceable message. It is usually very specific in contrast to authority which is much more general. I hope that helps!

authority is your ability to make rules. a mandate is a rule that people follow because of that authority.