Emergency situations are those that require immediate intervention. Someone could get hurt, injured, or worse if something is not done immediately.
Examples: a drunk friend walking out to his car with keys in hand, a drunk couple heading into a bedroom at a party, or a fight brewing between two people.
Non-emergency situations are those that may not require immediate intervention, but you still need to do something.
Examples: a friend is having academic trouble, is depressed, or may have an eating disorder.
Non-emergencies can turn into emergencies very quickly. For example, a friend who appears depressed may not seem like an emergency, but without appropriate and rapid help, might become suicidal. It is always best to intervene early!