Plato contends we are all made of the same three parts yet not all have the parts aligned in a healthy balance. The result is that greed, ambition, and foolishness rule in these unbalanced people. Plato lived through the democratic period in Athens' government and through the oligarchy period when the conquering Spartans installed the wealthy oligarchists as rulers of Athens, a move that unleashed a fierce retribution of bloodshed upon the unseated democratic rulers.
Plato rejected the rule of the mistake prone and seemingly unreasoning democratic faction and equally rejected the oligarchic rule of the retaliatory wealthy elite. After a period of seclusion, Plato wrote the Republic. In it he describes human nature and uses human nature (as he described it) as a metaphor and template for a reasonable government.
He assigns ruling authority to those who have a functioning alignment and balance between their three constituent parts and a dominant dedication to the highest: (1: lowest) love of money (laboring and merchants classes), (2: middle-most class) love of honor (military), and (3: highest) love of wisdom ("scientists, scholars, high-level experts, and similar sophisticates" [Jorn K. Bramann]).
His idea is that the two models he has seen don't work, so a third is needed. That third model is to make a government out of those who have the best minds by virtue of being best trained, best informed and best balanced (in the quote below, take note of and understand the "or"):
Until philosophers are kings, or the kings and princes of this world have the spirit and power of philosophy, ... cities will never have rest from their evils. (Republic)
[Jorn K. Bramann. Educating Rita and Other Philosophical Movies. Frostburg State University, MD.]
We have to put Plato in context. He lived as a student of Socrates in the city of Athens right after the Peloponnesian Wars--a disastrous time. He watched a jury condemn Socrates and saw the tumultuous politics of the time which had seen a number of would-be tyrants trying to maneuver their way to positions of power. It makes sense, I think, in his shoes to blame the system, a form of democracy that chose people for government service by drawing lots, which of course made no distinction for competence, virtue, or any other qualification. Plato saw this as counterproductive, and had faith that true philosophers would govern the city wisely and virtuously. We take for granted that democracy, or at least a democractic system, is the best form of government. Someone living during Plato's time might have seen it quite differently.
I agree with Post #3 that Plato did have something of what we would think of as elitist views. Plato felt that the best government would be one that was run by those who are naturally more gifted than the rest. I think that Plato would have liked the society in Brave New World to some degree. That is because that society was at least trying to have a situation in which everyone played the role in society for which they were best suited.
Plato envisions a system that was similar to that. He envisions a system in which the people with the greatest abilities rule over the others and the others pretty much accept the fact that they are not cut out for ruling on their own.
Plato favored an aristocracy ruled by philosopher kings. He did not believe most societies could maintain it though.
Plato’s aristocracy was based on merit. The leaders were strong, superior people who were well-educated and had the knowledge to rule.
We are thus led on to the conception of a higher State, in which … "kings are philosophers" and "philosophers are kings;" and there is another and higher education…
Plato also believed that aristocracies would devolve into timocracy (rule by property owners) when less superior people are admitted to the aristocratic class. That would further lead to oligarchy (rule by a few) where the rich rule over the poor. That would lead to democracy (rule by the many), which would inevitably reduce itself to tyranny (rule by one) when someone has to take power.
The text is here: http://classics.mit.edu/Plato/republic.mb.txt