Carol Marcus met James Kirk when he first got his Captain's Command. They stayed together for three months before he left. She found that she was pregnant afterwards and decided not to tell Kirk to keep her son from wanting to be a Starfleet Officer like his father. Kirk never found out until his son David Marcus was full grown.
David would become a scientist like his mother and he would work with her while she was building the Genisis Project. The Genesis project would spell the death to all the scientists except Carol as many were killed by Khan Noonien Singh and one was killed by Clark Terrill under the control of Khan. Lastly her own son was killed by at the Command of Kruge who was trying to steal the devise.
Carol was created for the second Star Trek film Star Trek II the Wrath of Khan. She was played by Bibi Besch.
After the death of her son, Carol established the David Marcus Research Center to find a way to solve hunger on all planets.
In the alternate time stream that follows the 2009 Star Trek film, Carol Marcus is a Starfleet officer. Her character, played by Alice Eve, is introduced in the film Star Trek: Into Darkness. Carol is the daughter of Admiral Alex Marcus.
She is assigned to the Enterprise as a science officer and continues in that capacity in the IDW comics series, beginning with the After Darkness story arc.