Final answer:
The quote refers to the guarantee by the U.S. federal government of a republican form of government in every state, as mandated by Article IV, Section 4 of the U.S. Constitution.
Step-by-step explanation:
The quote 'the United States shall guarantee ...a republican form of government' refers to the guarantee of a republican form of government in the United States. Specifically, this is outlined in Article IV Section 4 of the U.S. Constitution, known as the Guarantee Clause. It asserts that the United States federal government must ensure that each state maintains a government that is representative, where officials are elected by the people, rather than a monarchy, which is government ruled by a single person, or a socialist government, which could imply a range of governmental structures where the state has ownership and control of the means of production.
The notion of a republican form of government is one where the government's power is derived either directly or indirectly from the people, typically through elected representatives. This is further supported by the system's checks and balances, such as the election of the House of Representatives, indirect selection of the Senate and the President, and the judiciary appointed for a limited term or during good behavior. The federal government's commitment to defending the states against foreign invasion and domestic violence underscores the importance of this provision.