There are two possibilities. One is that Barclay's will get to keep the customers, and switch them to another credit card that they offer. This is what US Bank did to holders of the Northwest credit card after it merged with Delta. The other possibility is that AA will get to keep the customers, and Barclay's will sell the card portfolio to Citi.

Which path they take depends on their contracts and what they negotiate. But in either case, they will give you warning, so if you don't approve of what they decide, you should have sufficient time to close the card.