The Text
The United States shall guarantee to every State in this Union a Republican Form of Government, and shall protect each of them against Invasion; and on Application of the Legislature, or of the Executive (when the Legislature can-not be convened) against domestic Violence.
The Meaning
This provision, known as the guarantee clause, is attributed to James Madison. It has not been widely interpreted, but scholars think it ensures that each state be run as a representative democracy, as opposed to a monarchy (run by a king or queen) or a dictatorship (where one individual or group of individuals controls the government). Courts however have been reluctant to specify what exactly a republican form of government means, leaving that decision exclusively to Congress.
The section also gives Congress the power (and obligation) to protect the states from an invasion by a foreign country, or from significant violent uprisings within each state. It authorizes the legislature of each state (or the executive, if the legislature cannot be assembled in time) to request federal help with riots or other violence.