She was born around the turn of the 23rd century. (TOS: "This Side of Paradise"; TAS: "Yesteryear") It was implied that she was a descendant of human author Sir Arthur Conan Doyle. (Star Trek VI: The Undiscovered Country)
Amanda met Sarek, the Vulcan Ambassador to Earth. The two later married, and she returned to Vulcan with Sarek. (TOS: "Amok Time") The two were in the city of ShiKahr where Amanda gave birth to her only son, Spock, in 2230. (TOS: "Journey to Babel"; TAS: "Yesteryear")
Around 2249, when Spock was considering completing the kolinahr discipline, which would have purged the last vestiges of his emotions, he expressed his wish that Amanda not consider his choice to live a life free from emotion as a reflection upon her. Amanda assured her son that no matter what he chose, she would always be proud of him. Spock ultimately decided to enroll in Starfleet Academy rather than the Vulcan Science Academy after the head minister of the Vulcan Council referred to Spock's mother as a "disadvantage".
In 2258, Vulcan was attacked by the Romulan mining vessel Narada. Attempting to preserve the knowledge of the Vulcan culture, Sarek, Amanda, and the Vulcan Council went into the Katric ark, making evacuation by transporter impossible. Spock beamed down to the surface and evacuated them from the ark so that they could be beamed aboard the USS Enterprise. As the transporter was initiated, the ledge beneath Amanda collapsed, preventing a viable transporter lock, and Amanda was lost.
James T. Kirk later used Amanda's death to encourage an emotional reaction from Spock by questioning whether he ever loved her (as an eyewitness to Spock's reaction to her death, he knew this was his best chance to successfully do so), thus forcing Spock to recognize that he had been emotionally compromised by her death and was thus unsuited to command the current mission against Nero. Spock and Sarek later spoke about Spock's inability to control his anger at Nero for causing Amanda's death, prompting Sarek, knowing that suppressing such strong emotion would've been the last thing she would want Spock to do, to admit to his son that he had genuinely loved his wife, helping Spock achieve a greater control of himself. (Star Trek)
Following Amanda's death, Spock gifted a Vulcan amulet she had owned, which was made of vokaya, to Nyota Uhura. After Spock ended their relationship, Uhura tried to return the amulet but Spock refused. Spock was later able to use the amulet to track Uhura and the captured Enterprise crewmembers on Altamid. (Star Trek Beyond)
Background information Edit
Amanda Grayson was portrayed by Winona Ryder.
Additional scenes featuring Amanda were shot for the film Star Trek, but cut before the final release. One such scene features Amanda and Sarek (predating Nero's incursion, Ryder and Ben Cross were actually playing Jane Wyatt and Mark Lenard's "prime" characters), together with the newborn Spock. Another involves Amanda arguing with Sarek about the boyhood Spock and how he should react to bullying, after he has been involved in a fight with another Vulcan boy. Though Amanda at one point admits she wants Spock to embrace being Vulcan, she is of the opinion he should, every now and then, do the "Human" thing by occasionally walking away from conflict. Frustrated with Sarek, she goes on to do precisely that.
The writers expected that, because Spock was to lose his mother later in the film, writing Amanda into a lot of the early scenes of the movie would increase the impact of that loss. According to Producer Damon Lindelof, however, the writing team actually found the opposite turned out to be true. ("Amanda and Sarek Argue After Spock Fights" audio commentary, Star Trek (Special Edition) DVD/Star Trek (Three disc Blu-ray)/Star Trek: The Compendium Blu-ray special features)
One reason why the writers decided to have Amanda die in the film Star Trek was that their depiction of the friendship between Kirk and Spock was inspired by that between Paul McCartney and John Lennon, both of whom suffered through the death of a parent in childhood (Kirk loses his father, George Kirk, in the film).  In the film's script but not the final version of the movie, Amanda's final words were to tell Spock, ""It's okay. To be scared." Spock's memory of her, immediately after her death, was referred to in the script as "the ghostly image of her – beautiful, holding, frozen in space – an image he'll be haunted by forever – as it fades away, evanescing like mist." 
Concerning Ryder's performance as Amanda Grayson, Spock actor Zachary Quinto once stated, "Winona brought a real tenderness to the role and she really highlights the juxtaposition between her heartfelt way of being and Sarek's logic." (X)