What is the difference between apple juice and cider?
Apple cider has two meanings, but they both start with raw, pressed juice from crushed apples.   

Soft apple cider (normally just called cider) is simply the pressed juice, bottled.  It is cloudy from suspended apple particles, and turns brown from the oxidation, much as apples themselves do when cut and exposed to air. Hard apple cider is an alcoholic beverage made by allowing soft apple cider to ferment.

Apple juice is essentially the squeezed juice from the apples that has been highly filtered to remove the suspended particulates.  It is usually pasteurized to have a longer shelf life.

UK Usage:  Per ElendilTheTall, in the UK, the alcoholic beverage is called cider, and the other products are called simple apple juice.

See also:  MA government article