You shall make a breastplate of judgment... (28:15)
What does the priest's breastplate have to do with judgment?
The answer can be derived from the last line of the instructions for making it:
"You shall put the "urim vetumim" into the breastplate of judgment, and Aharon shall bear the judgment of the children of Israel upon his heart before Hashem continually." (28:30)
And elsewhere we see the urim vetumim explicitly mentioned in connection with judgment:
"You [Moshe] shall place your glory on him [Yehoshua], so that the whole congregation of the children of Israel listens to him. And before Elazar the priest he will stand, and will ask him for the judgment of the urim before Hashem. By his mouth they will go out and by his mouth they will return - he and and the children of Israel with him, and all the congregation." (Bamidbar 27:20-21)
Almost certainly then, the "judgment" of the breastplate refers to the judgment of the urim vetumim which are contained within it. Aharon, by virtue of carrying the urim vetumim, carries the people's judgment with him wherever he goes.
But what exactly does the "judgment" of the urim vetumim consist of?
The urim vetumim were a means of communication from God to the Jewish people. Perhaps, one of the things God would tell the people would be how to resolve difficult legal cases. In Ezra 2:62 and Nechemiah 7:65, for example, it is expected that the urim vetumim would be used to determine whether certain individuals were priests, and thus allowed to eat "terumah". In these verses, the "judgment" of the urim vetumim apparently means judgment of specific court cases.
However, in the rest of Tanach we see urim vetumim used for a different purpose: to ask God for military advice, in particular whether or not to wage a certain battle. This is how the urim vetumim are apparently intended to be used by Shaul fighting against the Philistines (Shmuel Alef 28:6), by David under attack by Shaul (Shmuel Alef 23:9-13), and by the 11 tribes fighting together against Binyamin (Shoftim 20:23-29).
To that list we can add the previously mentioned verses from Bamidbar, in which Moshe is told to appoint Yehoshua as his successor. What is the relevance of urim vetumim to this appointment? Yehoshua's main task as leader was the conquest of the land of Canaan. Presumably, consulting the urim vetumim for military guidance was an inescapable part of the conquest. As God describes things, Yehoshua will visit Elazar, then Elazar's urim vetumim will provide advice, and finally the people will go out to battle (that's the meaning of the expression "going out and returning", see here). The military advice Yehoshua would receive is called "judgment".
But since when is military strategy called "judgment"? To answer this, remember that before the monarchy, Israel was ruled by "judges". A glance through Sefer Shoftim makes it seem that sitting in court was a minor part of their job description. Only a few of them are ever listed as judging the people. But every one of them is listed as fighting wars against Israel's enemies.
Why then are they called "judges", if their main role was military? The answer can be derived from a comparison between different conceptions of the judge's role.* According to one conception, judgment is basically an intellectual task. The judge is to overcome the complications and ambiguities of the situation before him, figuring out exactly what the facts of the case are and exactly which passage of law applies to it. Once the verdict is issued, the judge's task is finished. This conception is strongly rooted in Greek, and therefore in modern Western conceptions of the judge's role.
In Tanach we find a different conception: that the judge's task is not only to arrive upon a verdict, but to enforce it. Enforcing the morally required outcome is so central to the judge's role that the phrase "to judge" is used as a synonym for "to save", as in Shmuel Alef 24:15, Shmuel Bet 18:19, and Yeshayahu 59:11. In this conception, it is natural to expect judges to be not only experts but leaders, and natural for leaders with sufficient moral fiber to also be trusted as judges. Thus we find Israel's military leaders, who saved the nation from its enemies, being called "judges". History remembers only their performance in crucial battles, but their title equally reflects the judicial role which they filled on a more regular basis.
It seems that the "judgment" of the urim vetumim included both aspects of Biblical "justice". As seen in Ezra and Nechemiah, the urim vetumim provided answers to legal dilemmas. And as seen in the other sources, they were a crucial source of military strategy. Many Jewish sources attest to the idea that intellectual aptitude must be accompanied by concern for the moral order and the well-being of others. Through the urim vetumim, God set an personal example in regard to this concern, by combining legal insights with the practical advice the people needed in order to defeat their enemies.
*In this I follow chapter 5 of R' Eliezer Berkowitz's book of collected writings.