Expiration is what happens when a lease or contract — or a period of time — has ended. The expiration of your library card means you need to sign up for a new one.
You might notice that your container of yogurt is marked with a date — it's the expiration date, or the day the product is no longer fresh enough to be sold or eaten. Other things with expiration dates include magazine subscriptions, rental agreements, and drivers' licenses. All of these are a sort of contract that eventually ends, or expires. When you breathe out, that's technically also expiration, from the Latin expirare, "breathe out," or "breathe one's last breath."