Democracy v. Republic: Federalist Papers Explain
"In a pure democracy, citizens assemble and administer the government in person." Federalist Paper #10[paragraph 13]*
If you lived in a pure democratic city, all the citizens would gather to do the work of the city. If the city needed a traffic light, all the city's citizens would have to gather, decide that a light was needed, decide where to place the traffic light, then go and install the traffic light.
What if the city decided to hire a company to install the traffic light? Then the city is no longer a pure democracy. The company hired to install the traffic light represents the city's citizens. The citizens of the city decided where to install a traffic light, but the company is representing the citizens when it installs the light.
In a pure democracy, the entire city would gather to decide the fate of alleged criminals, run the jail, and become the probation officers for criminals. The logistics of such a system seem impossible to imagine. It might work in a small village, but as the village grew, the citizens would eventually hire someone to fulfill the role of sheriff. Once a sheriff is elected or appointed by the town, he or she represents the citizens as he performs his duties.
It is clear that a democracy must be geographically tiny. Only in a tiny democracy can all the citizens meet and administer the government. And common sense makes it clear that personal rights could be easily trampled by the majority of citizens.
"Democracy doesn't cure the harm caused by faction. Usually, the majority will feel a common passion. There is nothing to stop the desire to sacrifice the weaker party or obnoxious individual. Therefore, pure democracies are always turbulent. They don't secure personal or property rights. They have short lives and violent deaths. Some political theorists think that after people become politically equal, their possessions, their opinions, and their passions will also be equal. This doesn't happen." Federalist Paper #10[13]
Republic
"A republic and a democracy differ in two ways: First, in a republic, the citizens elect a small number of governmental delegates. Second, a republic can have more citizens living across a larger country." Federalist Paper #10[15]
It is obvious that we live in a republic. We elect people to represent us in the government. Some people in government are not elected, but those people are hired or appointed by people we elect. For instance, the President, who is elected, nominates, and the Senate, which is elected, confirms supreme court justices.
*The Federalist Papers Modern English Edition Two, Webster, 2008.