Plato is among the most famous critics of democracy. His criticism is relatively simple, but potentially devastating. It runs as follows. Politics aims at achieving justice, and so political policy must reflect the demands of justice. Only those who know what justice is and have the self-control to enact what justice requires are capable of doing politics properly. Alas, the average citizen is dumb and vicious. Hence Plato's conclusion is that democracy is a fundamentally corrupt form of politics; it is the rule of those who neither know nor care about justice. In The Republic, Plato's Socrates argues for a philosophical monarchy, the rule of the wise and virtuous.
I've discussed this sort of issue before; not to repeat myself too thoroughly, the basic points are:
(1) The argument of The Republic is not that the average citizen is dumb and vicious; this would make a just society simply impossible on Socrates's account, since he is quite clear that a just society requires that people in the main act wisely and justly -- not just the rulers, but the typical citizen -- and that the way to work toward a just society is to strive to be more just and wise oneself. The whole point of the Republic is that society and citizens reflect each other. And, indeed, what other position could possibly be consistent with the Socratic approach as Plato presents it?
(2) Plato's Socrates does not spontaneously argue for the kallipolis; even the kallipolis is what Socrates comes up with only after his first proposal for a just society is rejected.
(3) Plato's actual criticism of democracy is that it has the minimum possible defenses against injustice. It works well as long as everyone agrees about what is harmful and not harmful. But it has nothing to guarantee that people will agree about this; everyone is his own city and can make up his own standards. Philosophy is a kallipolis thing; traditions are timarchical; calculating profit is oligarchical; so what else is actually left to make democrats cohere into a society of any sort? Rhetorical manipulation. Thus Plato's problem with democracy is that it is a breeding ground for demagogues, and thus has nothing to prevent the slide into the rule of brute force. While it would still be too extreme, it would be more accurate to say that Plato thinks the problem with democracy is that democracy makes the average citizen dumb and vicious, by forcing him into a situation in which rhetoric has more power than justice, and sophistry than wisdom.
But Plato's criticism of democracy is in fact not unqualified; in the Republic he is only considering the pure case. It's worth pointing out the Athenian's comments in the Laws (693d):
There are two mother-constitutions, so to speak, which you could fairly say have given birth to all the others. Monarchy is the proper name for the first, and democracy for the second The former has been taken to extreme lengths by the Persians, the latter by my country; virtually all the others, as I said, are varieties of these two. It is absolutely vital for a political system to combine them, if (and this is of course the point of our advice, when we insist that no state formed without these two elements can be constituted properly) -- if it is to enjoy freedom and friendship applied with good judgment.
When he goes on to discuss how Athens became corrupt, he argues that the issue was that the Athenians lost respect for authority and law: resisting authority, they began resisting their traditions and the admonitions of their elders; resisting the admonition of their elders, they began to resist the laws themselves; resisting the laws, they came to resist even the moral principle underlying promise and pledge. A city needs freedom -- but it also needs unity and wisdom, and neither, Plato argues, can be had from democracy as such.
