Believe it or not, the English title The Republic may not actually be the most accurate translation of this dialogue's Greek title Politeia.
"Politeia" means something much closer to our word "regime," which actually makes a bit more sense. Considering that Plato's city is an aristocracy in which the state controls practically every part of life, calling it a "republic," with the democratic associations that word has, is definitely a stretch. Socrates is more interested in simply proposing and examining different types of governments— or regimes—to see which one best fits his philosophical ideals.
Why all this mix-up? Well, it's actually a testament to just how popular Plato's dialogue was, even way back in Ancient Rome. Cicero, a famous Roman philosopher and statesmen, was a big Plato fan—so much so that he wanted to write his own version of the Republic, which he called in Latin Res publica. See the connection? We've just inherited his version of the Greek title.
And in case you were wondering why a book mostly about justice doesn't mention justice anywhere in the title, it actually has a little-known subtitle: "on the just." So you can calm down.