Abishag was King David's nurse in his old age. David felt cold due to his age, so it was Abishag's duty to lie against him and keep him warm. Nevertheless, this was not an intimate relationship.
David's eldest surviving son Adonijah attempted to usurp the throne while David still lived, but the prophet Nathan, queen consort Bathsheba, and King David foiled Adonijah's plan and had Adonijah's younger half-brother and Bathsheba's son Solomon, who God loved, anointed King of Israel instead. Adonijah humbled himself before Solomon, and Solomon pardoned him.
However, Adonijah still sought the throne. Adonijah approached Bathsheba and asked her to ask Solomon to grant Abishag to Adonijah as a wife. Adonijah sought to strengthen his pretensions to the throne by marrying Solomon's former nurse and believed that Solomon would not refuse a request delivered by Bathsheba.
Bathsheba conveyed Adonijah's request to Solomon, but Solomon recognized the threat to the throne and had Adonijah executed.