We're sorry to say that just because you might be a big Law & Order fan, it doesn't mean you're going to have an easy time with Plato's understanding of justice. We often think of justice from the perspective of our own legal system, but Plato is interested in thinking about it as a philosophical principle that determines how we should behave. It doesn't necessarily matter if it's part of the legal system or not.
The whole problem of justice begins in the Republic when Socrates questions Cephalus's idea that justice means telling the truth and giving people back what belongs to them. Socrates imagines that if you borrowed a sword from one of your friends who then went bonkers, it would not be just to give the bonkers friend his sword back.
So, justice is a bit more complicated. So complicated, in fact, that Socrates decides they need to imagine a city just to make sense of it. Socrates compares a well-run city to a well-run person, thereby suggesting that his definition of justice will work for both big political structures and for regular old individuals.
It turns out that there are two big issues to solve: 1) what is justice, and 2) will acting justly make you happy? While there's no one definition of justice offered in the Republic—remember, it's a dialogue, not an essay—Socrates does conclude that justice is 1) doing what you're best suited to do and 2) minding your own business (433a-433b).
What Socrates means is that justice is about unity and order, since great harm comes from any kind of internal strife or external distraction. In this way, justice is related to his favorite Delphic saying, "know thyself," since self-knowledge is also a process of ordering yourself and knowing what you should do.
Because justice is about order and harmony, Socrates then goes on to claim that acting justly makes you happier. Using the example of a powerful and wealthy tyrant, Socrates explains that even if this guy has a lot of creature comforts, he's actually miserable, because without justice to order his life, he's always doubting himself, doubting other people, and fearing for his life.
Yeah, we'll stick to justice, thank you very much.