(covers information from several alternate timelines)
Amanda Grayson was a Human teacher from Earth born around the turn of the 23rd century. (TOS: "This Side of Paradise") During the early 2230s, Amanda met Sarek, the Vulcan Ambassador to Earth. The two later married, and she returned to Vulcan with Sarek. (TOS: "Amok Time") In later years, in describing his parents' relationship, Spock stated that his mother "considered herself a very fortunate Earth woman." (TOS: "The Corbomite Maneuver")
In 2230, Amanda gave birth to her only son, Spock.
It was not uncommon for Spock to mention his mother's origins. (TOS: "The Enterprise Incident") While under the influence of the polywater intoxication, Spock regretted that he "could never tell her [that he] loved her." (TOS: "The Naked Time") Spock once spoke of Amanda's fondness of reading the works of Lewis Carroll. She often read stories, such as Through the Looking-Glass, to Spock during his youth. (TAS: "Once Upon a Planet")
In an alternate timeline created by the death of Spock in 2237, Amanda and Sarek separated and she returned to Earth, where in transit, she was killed in a shuttle accident at Lunaport. (TAS: "Yesteryear")
During the Babel Conference of 2268, she accompanied her husband aboard the USS Enterprise, and helped Sarek and Spock to reconcile some of their differences. Spock wondered why his father would marry an emotional woman; Sarek replied it seemed the logical thing to do at the time, a comment that Amanda found quite charming. (TOS: "Journey to Babel")
During this journey, Captain Kirk was at a loss on how to properly refer to her, calling her "Mrs. Sarek." Amanda said that her married name was usually unpronounceable by Humans, although she could do it, "after a fashion, and with many years of practice." She said to simply call her "Amanda." (TOS: "Journey to Babel") In the Vulcan society, she was referred to as "the Lady Amanda." (TAS: "Yesteryear")
In 2286, Amanda helped her son to re-educate himself, after his death and rebirth on the Genesis Planet and fal-tor-pan rejoining. In particular, she tried to help Spock rediscover his Human side. (Star Trek IV: The Voyage Home)
The script of "Journey to Babel" includes a description of Amanda that reads, "She's in her late fifties and still a fascinating woman... straight, slim, humor and warmth still alive in her... and guts. She married a Vulcan and came to live on his world where her human-woman emotions had no place. She has accepted every bit of the unemotionalism Vulcan could dish out with no loss of her own warmth and human caring... but it has had to be buried inside, in deference to her husband's customs and world."
When it came time to cast "Journey to Babel", the role of Amanda Grayson was initially offered to Anne Baxter, who turned it down. (Star Trek Magazine issue 171, p. 37) Amanda was ultimately played by Jane Wyatt in "Journey to Babel" and Star Trek IV: The Voyage Home. The voice of Amanda was provided by Majel Barrett in "Yesteryear". The role of young Amanda in Star Trek V: The Final Frontier was played by Cynthia Blaise. In 2009's Star Trek, Winona Ryder, who played her alternate reality counterpart in the film, also appeared as "this" Amanda in a deleted scene set before the divergence.
Unlike the character of Sarek, Amanda Grayson was not included in Star Trek III: The Search for Spock and is absent from the resurrection of Spock that takes place in the conclusion of that film. Her exclusion from the movie is because the production staff couldn't find a way to feature her in the story without her presence seeming overly sentimental. Executive Producer Harve Bennett reckoned, "All she would have contributed was sympathy. The economy of the story was that Kirk and crew get Spock back. Family is secondary. That would have depreciated the moment when Spock says, 'Your name is Jim.' Then we'd have to cut to mother and she would say, 'Oh my God, he speaks!'" (The Making of the Trek Films, p. 46; Trek: The Unauthorized Story of the Movies, 3rd ed., pp. 87-88)
William Shatner originally intended for Amanda to feature more in Star Trek V than she actually does, wishing to explore her relationships with Sarek and Spock. At one story meeting during which Shatner voiced these interests, David Loughery was concerned about accounting for Amanda's influence on Sybok, though Harve Bennett replied, "There are solutions to that." (Captain's Log: William Shatner's Personal Account of the Making of Star Trek V: The Final Frontier, pp. 64 & 65)
Once, when Sarek actor Mark Lenard was asked where Amanda was in the Star Trek movies, he replied, "In the kitchen! Where else would a good Vulcan wife be?" (Cinefantastique, Vol. 27, No. 11/12, p. 103)
The novelization of TAS: "Yesteryear" (as published in Star Trek Log 1) describes Amanda in an image of her from circa 2237 (i.e. shortly before her death in that episode) as being pictured "in her early thirties." This roughly matches the fact that, in the novel Sarek, her year of birth is established as being 2202.
In the Crucible book The Fire and the Rose, her death is established similarly to her death in the alternate timeline in TAS: "Yesteryear". She dies in 2311 in a shuttle accident when returning from an art exhibition in Paris.
The TOS novel Ishmael cites Amanda Grayson as being a descendant of Aaron Stempel, a lead character on the real-world television series Here Come the Brides. On this show, the character of Aaron Stempel was played by Mark Lenard, the actor who played Sarek in Star Trek.