When there's an emergency, the stakes are high and you may need to act quickly. There's a tornado! Someone's collapsed! The cat's stuck in a tree! It's an emergency! Call 911!
You can see the word emerge in emergency, and an emergency suddenly emerges — it happens all of a sudden and can take you by surprise. Governments declare a "state of emergency" if there is a disaster that they can't handle, and they need support. If you hear sirens and see flashing lights, there's probably an emergency . . . unless you're in a dance club.