Hibernian can be used to describe anything related to Ireland, including an Irish citizen. You can show your Hibernian pride by wearing green on St. Patrick's Day!
Today, Hibernian is mostly used in very old names for places and organizations, like the Hibernian Orchestra and the Hibernian Football Club. It's the Latin term for "Irish," first used in Greek geographical writing about the island that would later be named Ireland. Hibernian is from Hibernia, "land of winter," not the most accurate description of a country with fairly mild winters, but possibly based on snowy conditions when ancient Greek explorers first visited.