And the Scrolls have foretold, of black wings in the cold, that when brothers wage war come unfurled! Alduin, bane of Kings, ancient shadow unbound, With a hunger to swallow the world!— Song of the Dragonborn
Alduin is the first Dragon ever born, the jewel of Akatosh and original the crown of creation, and so the most powerful and ancient of all the dragons. However, his rule of ancient Men and Mer was brutal and tyrannical, and his evil brought about the revolt against the Dragon cult that eventually laid dragons low as a race.
Alduin is the one resurrecting all of his buried bretheren, one at a time from their various ancient burial mounds. How he came to reappear in Tamriel is known only to the Greybeards, but his first known appearance was at Ulfric Stormcloak's abortive execution at the hands of the Imperial Army and Thalmor in Helgen in central Skyrim. It has been deduced that he was probably attempting to eliminate the nascent Dragonborn of prophecy, whom he had somehow discovered would also be present at this event - to be executed along with Ulfric. However, Alduin's interference with the event unwittingly prevented the death of the Dragonborn, as well as that of Ulfric, leading to his own undoing and that of the resurgent Dragon race at the very halls of Sovngard.
As to the Greybeards' knowledge of Alduin's arrival in present-day Skyrim: he was apparently hurtled forward through time, thousands of years, by the humans who developed the Dragonrend Thu'um, independently of their dragon allies, in an attempt to destroy him. They apparently never realized he would again surface one day. In the present day, Alduin is defeated by the Dragonborn at the halls of Sovngard with the assistance of the original heroes who banished Alduin from their time.
Alduin is tremendously powerful, moreso than any other dragon. He attacks with a viciously powerful fire or frost attack, and with physical attacks with his tail and jaws as with other more powerful dragons. Additionally, he employs a magical attack wherein burning boulders rain down on his foes, inflicting physical and fire damage. He is effectively impossible to defeat without the Thu'um 'Dragonrend', which was created to depose him as ruler of the world in ancient times by Nord heroes.