How does Twelve Angry Men show the meaning of justice?

In Twelve Angry Men, twelve jurors are asked to decide the fate of a young man who has been accused of murdering his father. In the beginning, all of the jurors (save one, Juror #8) think the boy is guilty. When Juror #8 votes not guilty in the preliminary vote, the other jurors demand to know why. Juror #8 says the purpose of a jury is to determine whether the boy is guilty. Since the verdict is dependent on the principle of "innocent until proven guilty," Juror #8 does not consider the boy guilty because he is not convinced of his guilt. Juror #8 also adds the following caveat:

There were eleven votes for "guilty." It's not easy for me to raise my hand and send a boy off to die without talking about it first.

Begrudgingly, the other men agree to talk, with varied degrees of willingness and participation. Throughout their discussion, we begin to see real dialogue and thought processes that were previously lacking. Many of the men start to look inside themselves to see if they have a reasonable doubt or not about the defendant's guilt. By showing us the process these jurors go through, Twelve Angry Men shows the audience how the process of justice is meant to protect the innocent, even if that can sometimes mean a guilty man gets off scot-free. Here's a quote from one of the jurors that demonstrates this:

Nine of us now seem to feel that the defendant is innocent, but we're just gambling on probabilities. We may be wrong. We may be trying to return a guilty man to the community. No one can really know. But we have a reasonable doubt, and this is a safeguard that has enormous value in our system. No jury can declare a man guilty unless it's sure. 

At the end of the play, most of the jurors except for perhaps three of them (who vote "not guilty" for reasons of their own) truly believe the boy has not been proven guilty beyond a reasonable doubt by the evidence and testimony. Through his insistence on the safeguard of reasonable doubt, Juror #8 shows what justice really means in the United States.