A republic, not a democracy
The slogan "a republic, not a democracy" refers to the difference between direct democracy and representative democracy. The distinction is between direct democracy in which everyone votes, and the majority rules; and a representative democracy in which executives are elected to rule the people.
The ancient Greeks had direct democracy, i.e., the people voted, and the decision of the majority prevailed. Some New England town meetings are like that. In some U.S. states, a proposition will go on the ballot and be decided by voters directly, but this is rare; see legislature.
Although the slogan is used by some to alert listeners to the particular kind of democracy which the United States Constitution provides, it is misleading because the phrase "not a democracy" implies that representative democracy is not a democracy at all - a notion which most political scientists would reject as nonsense.