An exception is something that is left out or not done on purpose. An exception to a rule does not follow that rule.
This word is used for all sorts of things that are not usual or usually allowed. The saying ”i before e except after c,” is about an exception to a spelling rule. If you run every day but take Saturdays off, you're making an exception. If your teacher punishes you for being late even though you have an excuse, she might say, "I'm sorry, but I can't make an exception." When you see exception, think "Something different than usual is happening."