Justice has to do with judging. It's general usage is that of correct judgement. You might see it as the middle ground between mercy and retaliation. Usually this kind of thing is handled by courts. In the US people are told "not to take justice into their own hands." because it's the job of the courts, police, etc.
There are other kinds of justice...for instance "poetic justice" is a term used when circumstances suddenly change that seem to give some form of justice, but not through anyone's hands directly. For instance, a bully in middle school picks on all the children and everyone is afraid of them, but then in high school he stops growing and becomes the short little kid that gets picked on. Some people would call that poetic justice (or irony, perhaps as well).
You'll often see poetic justice in films where they can't have the "good guy" kill the bad guy, as they want the good guy to be merciful but still have the bad guy die. So the good guy attempts to help the bad guy, but by some accident or betrayal, the bad guy dies anyway.
Justice is often represented by the concept of balance.