OK, a Dragonshard, in a nutshell, is a special new kind of gemstone, unique to the Eberron setting, that comes in three different types.
Siberys shards - named for the mythic dragon Siberys the Sky Dragon, yellowish-gold in color, with bright gold streaks in the shape of symbols (almost) that almost seem to move within the crystal. Siberys Shards enhance the special powers of beings who have been Dragonmarked (a whole other topic).
Eberron Shards - named for the mythic dragon Eberron, the Dragon Between, it's said that his body makes up the world itself. Reddish shards with crimson designs inside, they enhance magic spells.
Khyber Shards - named for the mythic dragon Khyber, the Dragon Below, which represents the underworld below eberron, these jet stones with dark markings inside have properties that make them good for binding elementals into them, and using that power to give objects elemental powers.
That's the five-minute explanation, but to learn more about them it would be helpful to take a peek in the Eberron Campaign setting book, where they have their own section.