What are the two jurisdictions/systems of courts in the United States? What are the roles of each system?
It's difficult to tell by your question which systems you are referring to, as our court system in the United States is immense and complicated. I am assuming you mean the Juvenile Court System, and adult court.
The Juvenile system, like the body of law that applies only to juveniles, deals with crimes committed by people under the age of 18 at the time. As a society, we have come to believe that people of younger ages are both less capable of telling right from wrong in some circumstances, and more redeemable in terms of "curing" their criminal behavior, or at least reforming it. It is also not wise to place minors into the criminal and prison systems with violent adults. So we have a completely separate system to deal with them.
Adult court, of course, deals with those criminals who broke the law over the age of 18, or those who are minors, but the court has determined they have committed an adult crime and should be tried as such. In criminology, conventional wisdom suggests that the adult criminal mind is more difficult to reform, and is perhaps criminal in behavior and thought patterns for different reasons than juveniles. So penalties and guidelines in adult court are generally more severe than that for juveniles.