Winona Kirk was the wife of Starfleet officer George Kirk and the mother of James T. Kirk. She was with her husband aboard the USS Kelvin in 2233 before it was destroyed; they named their son after her father James and his father Tiberius.
In 2233 of an alternate reality, the Kelvin fell under the attack of the Romulan mining vessel from the future called the Narada. Winona was in the final stages of pregnancy at the time, and was wheeled to the escape shuttles on order of her husband, who had assumed the duties of command following the death of Captain Richard Robau at the hands of Nero. She went into labor shortly before boarding Medical shuttle 37 and evacuating the ship. George Kirk spent his final moments on the comm with his wife, deciding on the name "James Tiberius" before the Kelvin collided with the Narada.
She may have a brother, since James T. Kirk said he was staying at his uncle's farm in Star Trek Generations. It was never made clear which of his parents had a sibling though, or if he was just using the term for a long time family friend. In Star Trek, an uncle Frank was cut from the script and reworked into Winona's new husband heard in the film.
In a commentary track on the Star Trek DVD, producer Damon Lindelof stated that, in an early form of the script, she died in childbirth - in an emergency, the transporter could be used to remove a baby from the womb, but at the cost of the mother's life. According to J.J. Abrams, earlier scripts also included her in the angry telephone call during young Kirk's joyride, but this was dropped since, given her portrayal in his birth scene, it would feel "counter-intuitive" to cast her in this negative light.
Information from several of the novels and short-stories states she was born in 2210 and was of Sioux descent.