There are several reasons. The first is due to one of water's many amazing properties: Aside from being necessary for all life (and refreshing on hot days), water is one of the few substances that is slightly denser as a liquid than as a solid. This is why ice cubes float in water.
Second, and slightly more interestingly: Most icebergs actually contain a lot of air. Far from being the solid blocks of ice many people imagine, icebergs are riddled with billions of tiny, trapped air bubbles, giving the huge bergs their white appearance.
Third, icebergs are made from fresh water. Because of the dissolved salts in ocean water, it is denser than freshwater, adding bouyancy to the icebergs.