"So long as I do not know what the just is, I shall hardly know whether it is a virtue or not and whether the one who has it is unhappy or happy." (354c)
Big Problem Alert. Socrates is pretty sure that if you don't know what justice means, you don't know squat. Without a basic definition of justice, Socrates can't begin to evaluate whether it's good or not.
"And this, then, is the genesis and being of justice: it is a mean between what is best—doing injustice without paying the penalty—and what is worst—suffering injustice without being able to avenge oneself." (359a)
Instead of attributing any moral value to justice, Glaucon's formulation presents it as a cowardly compromise for those afraid of being the victims of injustice. Glaucon is hopeful, and so are we, that Socrates can find a good argument against this definition.
"...no one is willingly just but only when compelled to be so." (360c)
In another bleak assessment of justice, Glaucon again presents himself as the devil's advocate, coming up with a very negative idea to provoke Socrates into a persuasive defense. Playing the devil's advocate is a good way to get someone (or yourself) to consider objections to a point you're making; it can help you cover all the bases and make sure your reasoning is secure.
"If we should watch a city coming into being in speech... would we also see its justice coming into being, and its injustice?" (369a)
In this famous Eureka! moment, Socrates explicitly lays out the primary goal of imagining a city: to discover the meaning and origin of justice.
"In the next place, get yourself an adequate light and look yourself... whether we can somehow see where the justice [in the city] might be and where the injustice..." (427d)
Using the metaphor of light and exploration, Socrates imagines that he and his philosophical pals need to search through their city looking for that elusive thing—justice.
"Well, then, my friend... this—the practice of minding one's own business—when it comes into being in a certain way, is probably justice." (433b)
Socrates imagines that justice is a principle not so much of fairness but of just doing your own thing. Did you see that one coming? Of course, by "doing your own thing," Socrates doesn't mean that you should just go out and do whatever you want; he means that everyone has some innate qualities or "business" that they should develop and put to unhindered use.
"Then the just man will not be any different from the just city with respect to the form itself of justice, but will be like it." (435b)
This is Socrates's famous formulation of how his imaginary city is a mirror image of the individual. Like city, like person. Got it?
"Do you suppose it is anything surprising.... if a man... looks quite ridiculous when... he is compelled in courts or elsewhere to contest about the shadows of the just...?" (517d)
Socrates is worried about the fact that philosophers can sometimes come across as a little bit nutty. As he explains here, it's hard for philosophers to seem "normal" because the truths they are trying to communicate can sometimes be hard to communicate and hard for others to accept.
"Too much freedom seems to change into nothing but too much slavery..." (564a)
It's exactly this relationship that makes Socrates so suspicious of democracy—and that makes him worry that justice and freedom aren't as connected as we think. Why does too much freedom lead to slavery? Can the majority actually be trusted?
"Shall we hire a herald then... or shall I myself announce that... the best and most just man is happiest..." (580b-c)
Showing a sense of humor, Socrates jokes about just how significant this final conclusion is. With or without a herald to announce it, anyone who's made it through the whole of the Republic will recognize what an achievement it is to have definitively concluded that being just makes you happy.