The definition of a republic is a society in which the citizens rule through elected officials. A republic is also absent of any sort of ruling party that usurps power through birthright.
Although many people think of the United States as a democracy, it is actually a democratic republic. In other words, the United States draws elements of both governments. The republic aspect of government is the officials that were elected by the citizens to represent them in Washington. The democratic aspect of the government is in the selection process of those officials, in which the majority of the citizens decide who those elected representatives are.