If you don't have Apple Care and your one year is up, you'd contact American Express. They may have you get a repair estimate, and then they'd decide whether to pay for the repair, replace the item, or reimburse you for the original cost. (You would need to have all of your original receipts and not be behind in paying your bill at the time you filed the claim.) More info can be found here:
https://www124.americanexpress.com/cards/cda/dynamic.jsp?name=BuyersAssuranceCCSGFAQ&type=intFAQ
Now, what happens if you buy Apple Care? (I purchased my MacBook with AmEx and bought Apple Care at the time, so I wondered that myself.)
This is what the Buyer's Assurance Plan says (on page 2):
"If you purchase an additional service contract or extended warranty with a product which is otherwise eligible under the Buyer's Assurance Plan, and the combined coverage provided by both the original manufacturer’s warranty and the purchased service contract does not exceed five years, then the product is eligible for coverage under the Buyer's Assurance Plan. The Buyer's Assurance Plan will extend the warranty time period and mirror coverage of the original manufacturers' warranty up to one additional year after both the original manufacturer’s warranty and the purchased service contract have expired. ... "
Since the Apple warranty is one year and Apple Care adds three years, the Buyer's Assurance would be tacked on when the Apple Care ended. However, it's not really clear from the way that is worded whether or not you'd have to buy the extended warranty at the time you bought the product. That would be a good question to ask American Express.
The Buyer's Assurance Plan also says (page 2):
"If you buy an additional service contract or an extended warranty for a computer, computer component or part that already comes with an original U.S. Manufacturer’s warranty, unless such coverage is provided from and
administered by the original manufacturer, coverage under the Buyer’s Assurance Plan does not apply."
Since Apple Care is provided by Apple, it would seem that this exception would not apply. That would be another question that might be worth asking American Express.
The American Express Buyer's Assurance plan can be found at
https://www124.americanexpress.com/cards/benefits/includes/pdf/buyersAssurance20 03.pdf