Henry FitzRoy is the bastard son of King Henry VIII and one of his early mistresses, Elizabeth Blount. FitzRoy was the first son of Henry's who lived beyond infancy, and the only illegitimate child that the King officially recognized as his own. FitzRoy is born in episode 1.02 of The Tudors; about five or six years later he is given noble titles, becoming Duke of Richmond (though Henry cannot name him as his heir over his legitimate daughter Mary). However, FitzRoy dies of the sweating sickness in episode 1.05, to Henry and Elizabeth's grief.
In the first episode of The Tudors, not long after ending her sexual relationship with the King, Elizabeth Blount- one of Queen Catherine's ladies-in-waiting- discovers she is pregnant with Henry's child. She secretly informs the King's first minister Cardinal Wolsey, who informs Henry and tells Elizabeth he will deal with her husband and orders her to go into seclusion for the remainder of her pregnancy; it is up to the King whether or not he will acknowledge the child. In the next episode, Elizabeth gives birth to a healthy son; Henry is ecstatic, since his marriage to Catherine has only produced a daughter and several stillborn/ miscarried children. Now that Henry has proven himself capable of siring surviving male children, he can conveniently blame Catherine for his lack of a male heir, though Catherine has constantly prayed for a son. Elizabeth is very relieved that Henry will provide for them and recognize his child, but she is banished from court, and Catherine of Aragon (unsurprisingly) treats her and her son with hostility.
Henry FitzRoy is presented to Henry at a ceremony in episode 1.05, watched tearfully by his mother. Henry kisses and embraces his young son, then invests him in the regalia of a nobleman, naming him Duke of Richmond. Catherine is furious at this action, as FitzRoy's elevation threatens the position of her legitimate daughter, Princess Mary; she incorrectly blames Cardinal Wolsey for influencing the King to advance his bastard son over his daughter. However, FitzRoy does not prove to be a threat, for at the end of the episode he falls ill with the sweating sickness and dies despite the efforts of the royal physicians. Elizabeth, dressed in mourning clothes, sobs over her son's deathbed; elsewhere, Henry quietly cries as he stares at the miniature crown and dirk he gave his son at his investment as Duke.
FitzRoy did indeed die of the sweating sickness, but in actuality he lived until age 16; he was able to take up his role as a Duke, and even had a tentative betrothal to another noble house at the time of his death. FitzRoy was also present at the execution of his father's second wife, Anne Boleyn. When Henry's second wife was failing to bear sons, King Henry once again seriously considered having Henry Fitzroy established as Crown Prince.
Interestingly, the Duchy of Richmond (formerly the earldom) was the domain of Henry Tudor (Henry VIII's father and Henry FitzRoy's grandfather) prior to his ascent to the throne. Henry Tudor became King following his defeat of King Richard III and his subsequent marriage to Elizabeth of York.