This is the 13016th most frequent Russian word.
любимой
"Любимой" means "beloved" or "darling" (feminine form).
Here, 'любимой' is in the dative case, modifying 'учительнице' (teacher), indicating the recipient of the action.
Она подарила цветы любимой учительнице.
She gave flowers to her favorite teacher.
In this context, 'любимой' is in the instrumental case, modifying 'девушкой' (girl), used after the preposition 'с' (with) to introduce someone who is a significant or beloved person.
Вчера мы ужинали с его любимой девушкой.
Yesterday, we had dinner with his beloved girlfriend.
In this sentence, 'любимой' refers to the feminine form in the instrumental case, modifying 'писательницей' (author), indicating possession or association.