Criminal cases involving minors (usually defined as people younger than 18) are typically called juvenile delinquency proceedings. The juvenile justice process differs greatly from its adult counterpart, such as when it comes to the trier of fact: The accused isn't entitled to a trial by jury in most states. (See Do juveniles have a right to trial by jury?)
But juveniles share at least one important right with adults: They are generally entitled to the government proving guilt beyond a reasonable doubt. (For information on exceptions to this rule, as well as information on jury trials in juvenile cases, see Constitutional Rights in Juvenile Cases.)
For more on all aspects of juvenile cases, see The Juvenile Justice System.