A federal republic government is divided between two levels of independent sovereignty, the national government and the states (or provinces, in some countries). Unlike a unitary system, the states or analogous subdivision in a federal government are not dependencies of the national authority. A federal republic does not have a monarchy but rather some form of representative government.
Wikipedia describes a federal republic as a "federation of states" with a republican form of government. Republicanism is a vague term that encompasses many different concepts, but the underlying principle is that the people are the ultimate source of political power. Thus, the people in a federal republic typically exercise rights of citizenship like voting and holding public office.
National Paralegal College expounds on the difference between the power of the national government and state governments in a federal republic. Whereas the federal government's powers are limited to those enumerated within the text of the Constitution, the states' powers are only restricted insofar as they may not pass laws that violate the Constitution.
Cornell University's Legal Information Institute explains that under a federal system, the national and state governments have jurisdiction over different spheres of public policy, although there is sometimes overlap. For example, the federal government normally takes authority in affairs that affect the entire country: national security, international relations and coining money. State laws typically deal with day-to-day issues that people deal with: crime, marriage, education and traffic regulation.