A birthday was the anniversary of someone's birth. Birthdays were celebrated by many cultures.
River Song was taken by the Doctor for her birthday to the last Winter Frost fair on the River Thames in 1814. Stevie Wonder unknowingly performed before being taken back to his original time. (TV: A Good Man Goes to War)
The Doctor had long forgotten his birthday by the time he had reached his twelfth incarnation, but the Master remembered it from their childhood friendship. As Missy, the Master created an army of Cybermen from deceased humans and gave control of the army to the Doctor as a gift with which he could potentially rule the universe. (TV: Death in Heaven)
Princess Clio was poisoned by her Uncle Ferdand on her fifth birthday. After the First Doctor put her into stasis, she was only awoken on subsequent birthdays, each of which was attended by an incarnation of the Doctor. (PROSE: The Glass Princess)