I have an advanced degree in philosophy and ethics, so I feel compelled to correct a few things here. The Republic is not a dialogue between Plato and Socrates. It is, rather, Plato using Socrates as a character through which he expresses his own philosophies. The book is also not a record Plato kept of Socrates' dialogues. It's rather widely accepted that Plato's critiques of democracy found within The Republic were influenced, in part, by the death of his mentor, Socrates, at the hands of a democracy.
These things said, this book reveals a great many problems that can arise in attempting to build the best society, not the least of which is that it can only be done when the individual builds that society first within him or herself. It's a foundational piece of literature that should be read early and revisited often to fully grasp the weight of the work. In today's society, it may be more important than ever that we look to these critiques of democracy so that we may attempt avoid the pitfalls Plato described.