A representative democracy gives the decision making ability to elected officials, but maintains control through the right to vote. Publius explained that the Constitution established a representative government not a democracy. He also stated that maintaining a true republic would be difficult and require a constant watch over the balance of powers between the national and state governments by the citizens. The papers stressed that the states are to retain powers that the Federal Government cannot change or affect, and even though today our states seem to be almost useless appendages in the Federal Government, they were to be essential parts of the Constitution.
There are those that believe it is the government's place to create eternal peace and harmony among its citizens. Madison understood that faction is natural and a consequence of liberty. He worked at creating a government that kept this in mind to benefit society, and a republic can best control groups from having too much control. Republics historically had failed and Madison was convinced it would work in America because of its size. With larger numbers it would be much harder for specific groups to gain enough power to make real changes, and actually, time has shown the difficulty in any one party maintaining a permanent governing majority.
Federalist Papers #52 - 77 explain how the seats of power are formed and operate as well as what powers they have and how they are checked and balanced. The House of Representatives displays a more republican approach with a representation of the people, whereas the Senate provides an equal representation of each state, bringing federalism into play (however, this has changed with the addition of the 17th amendment). Madison also pointed out that the leaders would have to play by all the same rules as everyday citizens. Unlike republics of history they would receive no special treatment. Although, I think they do receive special treatment - consider how they do no have to participate in Obama Care.
Finally, in order for the Constitution to work as the founders planned, it was assumed the people would always be involved in public debate and understand what was going on in order to hold their leaders accountable. One can only hope that the general population will once again become involved and do their research and truly understand what is going on instead of simply going by what sounds or looks good.