You've probably encountered the line somewhere. One party to a discussion expounds on a political point and laments that the majority rules and that is what "democracy" is all about. The respondent sneers back, "But we are not a democracy, we're a republic."
Which is more or less true. The problem is understanding what the difference is.
Majority rule in a group is one way of making decisions. It insures that most members are satisfied. Unfortunately it doesn't take a very deep analysis to realize that simply because most of the members want something doesn't mean the decision is right or just. I'll ask a class to vote on whether or not we should continue the lesson or take the rest of the day off. Even though the majority might prefer heading to the coffee shop, that isn't the most beneficial choice for achieving course objectives. Majority rule and democracy aren't necessarily good.
The founding fathers knew that.
Indirect or representative democracy is an alternative. It allows for greater expertise in the decision making because chosen leaders can familiarize themselves with the issues. They can evaluate in terms of priorities. They can allocate the dollars where needed not where wanted by the loudest voices. But is that a republican (small "r") form of government. Not quite.
Just as many textbooks offer different definitions of politics, so also you can find many definitions of a republic. The book my school is currently using offers possibly the weakest definition I've ever seen:
Well, that might work for some purposes, but it doesn't satisfy me with regard to the form of government which the Constitution of 1787 created for the United States. It was actually a bit farther removed from "representative democracy" and the citizens input.
The Founding Fathers were wise enough to clearly understand the dangers of majority ruling. They wanted a government that could respond to the majority yet not be controlled by the masses. They wanted a structure which was linked to the electorate but not overwhelmed by it. That's the sort of republic I like to reference in classes.
A republic is a form of government in which representation is chosen by a process that may be one or more steps away from the electorate.
Take a fresh look at the Constitution and notice how seldom the citizenry is voting for federal government officials. As originally written, only the weakest members of the government serving the shortest terms were chosen by direct election. It is only the House of Representatives which is elected by popular vote.
All other components of the government were chosen by a multi-step process which isolated the choices from popular emotions.
The senior chamber of the legislature, the Senate was appointed by the legislature of the state which they represented. The state legislature was elected but the choice for senators was then out of the hands of the citizens. This insured senior, mature, experienced, capable decision-makers would serve as a brake on the emotions of policy. The system prevailed until ratification of the 17th Amendment in 1913. That is 125 years of appointments rather than elections.
The President was chosen by an Electoral College. No mention is made of a citizenry casting votes. That little nicety isn't required. The electors are chosen by the legislatures of the states using whatever method they select. There isn't even consistency in that process. We have democratized the system with elections, but the only Constitutional requirement is that the state legislatures appoint some electors who then assemble to chose the leader. Face-to-face assembly isn't even required in that activity.
The judiciary of the United States is even further isolated from the citizens. All federal judges, not simply Supreme Court justices, are appointed by the President (who isn't required to be elected by citizens) and confirmed by the Senate (which wasn't elected either before 1913.)
Over the 225 years of our Republic we have shifted heavily toward democracy and given the citizenry a much more powerful voice. The result is what we have today and a very good case might be made that the worst fears of the Founders are being realized. The majority is increasingly ruling, for better or worse. The government is increasingly driven to pander to that majority without regard to justice or even common sense. The kids get ice cream and cookies every day and never have to eat their broccoli.