An excuse is an explanation for something that went wrong. When we give an excuse, we're trying to get someone to cut us some slack.
When you finish dinner and ask, "May I be excused?" you've used one of many meanings of this word. It’s generally a reason for behavior: A note from your parents saying you have a doctor’s appointment can excuse you from school for the day. If your dog is sick, he has an excuse for barfing in the house. But be careful — making excuses can get awfully close to whining.