Wikipedia
thumb|right Abishag (Hebrew Avishag אבישג) was a young woman of Shunem, distinguished for her beauty. She was chosen to be a helper and servant to David in his old age. Among Abishag's duties was to lie next to David and keep him warm ("they put covers on him, but he could not get warm"); however, David did not have sexual relations with her (1 Kings 1:4).
The Interpreter's Bible notes that
the Hebrews...believed that the fertility of the soil and the general prosperity of the people were bound up with the fertility of the king. David by this time was old and decrepit and his sexual vigor is called into question. Attempts are made to remedy the situation. The first cure is to heap clothes upon his bed in order to secure such physical heat as might render him capable. When this fails a search is made for the most beautiful woman in the land. Great emphasis is placed upon her [Abishag's] charms. The LXX supports this by translating in vs. 2, "and let her excite him and lie with him." The fact that the king did not have intercourse with her is decisive in the story. If David was impotent he could no longer be king.
After David's death Adonijah (David's fourth son), persuaded Bathsheba, Solomon's mother, to entreat the king to permit him to marry Abishag. Solomon suspected in this request an aspiration to the throne, and therefore caused Adonijah to be put to death (1 Kings 2:17–25). (In the earlier story of Absalom's rebellion, it is noted that having sex with the former king's concubine is a way of proclaiming oneself to be the new king.) He may have married her, at the pressure of his mother.
Some scholars point to the possibility that Abishag is the female protagonist in the Song of Songs.