The definition of a republic in the minds of the founding fathers was, "a guarantee of liberty for the people offset by limited power of government". They felt it was essential for these two forces to be kept in balance for the stability of the political system. Too much liberty leads to a corrupt government, while too much government destroys liberty.
The founders were afraid of a pure democracy because giving everyone the right to vote would allow a majority to trample on the rights of the majority. This would occur when evil men incited the people by harnessing their emotions. To avoid this problem, certain controls were removed from the people.
The president and members of the senate were selected by officials of each state and not by the people. In addition, those without property could not vote. The property requirement was relaxed over time so that by 1850, all white male citizens could vote. Popular vote for senators was not established until 1910.
Both of these controls were taken from the Roman republican model.