Alpine snowboarding is the practice of turning by carving the snowboard (such that the board turns by using the radius sidecut of the edge), as opposed to skidding the snowboard (where the board is traveling in a different direction than it is pointing). Both traditional snowboard racers (though not necessarily boardercross racers) and recreational carvers are alpine snowboarders.
Alpine riders use hard plastic snowboarding boots, which resemble ski boots, except that they tend to be less stiff in the ankles and have a shortened heel, to minimize hanging over the edge of the snowboard. They tend to angle their feet much more forward than other snowboarders, and also ride narrower boards.
Alpine boards are usually, but not always, longer and much stiffer than freeride boards, as the particular demands of carving usually require as much usable edge length as possible. The hard plastic boots stiffens the ankle joint up significantly, making it more difficult to make small ankle adjustments while making skid turns, but making the board much more stable and powerful at higher speeds and the much higher g-forces typically felt by an alpine snowboarder in carved turns.
An analogy made by some alpine enthusiasts is that freeride and freestyle snowboards are like dirt bikes, and alpine/carving snowboards are like road bikes. (Hence riding a freestyle snowboard on groomed slopes is like riding a dirtbike on a road track or what is called SuperMoto.)