Overview
A Dragonborn, Dovahkiin which can mean both "dragon-born" and "dragonkind-hunter-born", is a rare individual who has been born with the blood and soul of a dragon, but the body of a mortal. They can naturally learn and speak the ancient and powerful tongue of the Dov, called the Thu'um.
Dragonborn scarcely exist, and it is rare for more than one to appear at the same time. They can belong to any race or gender, and the dragon blood is a gift bestowed to certain mortals favored by Akatosh. Despite their divine souls, Dragonborn possess a mortal free will and may serve as champions for either the Nine Divines or the Daedric Princes. It is unknown if the Daedra claim the souls of their Dragonborn champions as they do with their other servants. Their mortal bodies also subject them to mortal vulnerabilities; old age, death, and disease all affect Dragonborn as they do common mortals.
Some Dragonborn individuals have knowledge of the Thu'um and can consume a slain dragon's soul and absorb its knowledge directly, allowing them to learn Words of Power instantaneously, and then use the power of a slain dragon's soul to unlock that Shout. It is said that the Blades have always guided, protected, and served the Dragonborn, whom they consider "The Ultimate Dragon Slayer." The Greybeards, however, see the Blades as ambitious assassins who seek to mold the Dragonborn into a weapon for their own ends.
Prophecy
The Prophecy of the Dragonborn, generally attributed to the Elder Scrolls, and sometimes to the ancient Akaviri, tells the following:
When misrule takes its place at the eight corners of the world
When the Brass Tower walks and Time is reshaped
When the thrice-blessed fail and the Red Tower trembles
When the Dragonborn Ruler loses his throne, and the White Tower falls
When the Snow Tower lies sundered, kingless, bleeding
The World-Eater wakes, and the Wheel turns upon the Last Dragonborn.
Dragonborn In History
The first Dragonborn in history was Miraak, a Dragon Priest from the Merethic Era. Instead of using his Voice to fight Alduin, Miraak decided to serve the Daedric Prince Hermaeus Mora instead, making himself an enemy of the Dragon Cult he once served. He was eventually defeated by the dragons and hid in Apocrypha. He was later encountered in the Fourth Era and did battle with the Last Dragonborn before being destroyed by Hermaeus Mora.
The next Dragonborn was purportedly Saint Alessia, the Nedic slave-queen who was blessed by Akatosh and given the Amulet of Kings to seal the covenant between them. This began the tradition of all legitimate rulers of the Cyrodilic Empire referred to as Dragonborn, as only they could wear the Amulet of Kings and light the Dragonfires in the Temple of the One.
Reman Cyrodiil was considered to be the next Dragonborn after Alessia. Upon their defeat, the Akaviri invaders recognized him as Dragonborn and swore unceasing loyalty to him.
Tiber Septim, founder of the Septim Dynasty of Tamriel, was the next individual to be recognized as Dragonborn. His heirs continued the tradition and were all considered Dragonborn Emperors. This lineage ended with the death of Martin Septim during the Oblivion Crisis.
The Last Dragonborn, otherwise known as the Laat Dovahkiin, appeared during the Fourth Era to fulfill the Prophecy of the Dragonborn. His purpose was to prevent Alduin the World-Eater from consuming all of Mundus and the souls of Sovngarde when the dragon appeared from a temporal warp known as the Time Wound, after having been sent into the future during the Dragon War. The Last Dragonborn also later defeated the first Dragonborn, Miraak, in Apocrypha after dealing with Hermaeus Mora in order to weaken Mora's disruptive champion.
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Source: http://elderscrolls.wikia.com/wiki/Dragonborn_(Lore)