Minor skin injuries such as cuts or tears (lacerations) or minor scrapes and abrasions don't require a trip to the emergency department. In fact, they probably don't even rate a call to your doctor.
Cuts and scrapes that are smaller than a half-inch (1.3 centimeters) in length, aren't on the face, don't go below the skin (no fat or muscle tissue is injured) and stop bleeding in less than 10 minutes (with pressure) are considered superficial wounds. Superficial wounds can be treated at home. First, apply pressure to stop the bleeding. Then, clean and bandage the area. That's it. If you have a standard first-aid kit in your home, you're already equipped to deal with superficial wounds without needing a medical professional.
However, if your cut is bleeding profusely, spurting blood, or continues bleeding nonstop after 10 to 15 minutes of firm pressure to the area, it qualifies as an emergency. Of course, any lacerations that are large or deep, that can't be cleaned or that don't appear to be healing after a few days should be evaluated by a doctor.