Republic is a word used to describe a state or nation that elects representatives to govern themselves.
“The United States is a republic because we vote every year on new representatives to go to Washington, D.C.”
Republican is an adjective used to describe a nation that elects representatives. It is also a name of a political party in the United States.
“Many nations have recently gotten rid of their dictator leaders and are thinking of adopting a republican form of government.”