Apple cider can refer to two things:
First, apple cider (also known as sweet cider) is similar to apple juice in that it is made from pressing fresh apples. Unlike apple juice, apple cider is less filtered and typically unpasteurized, giving the cider a deeper, richer flavor. Apple cider is typically a seasonal product, and is used in the same ways as apple juice.
Secondly, apple cider can refer to hard cider, which is apple juice that has been fermented to make it alcoholic, and typically carbonated. While there are mass produced brands of hard cider available in the United States market (many of them made by large beer conglomerates, and tasting of Jolly Ranchers), quality hard cider is making a resurgence -- look for producers like the Wandering Aengus Ciderworks from Oregon or Eric Bordelet from the Normandy region of France. These are much more complex, and typically drier than commercially made hard ciders.