This is the 10226th most frequent Russian word.
ближайшую
"Ближайшую" means "nearest" or "closest" in English.
The word 'ближайшую' is the accusative feminine singular form of 'ближайший,' meaning 'nearest.' It modifies 'библиотеку' (library) in this sentence.
Мы пошли в ближайшую библиотеку.
We went to the nearest library.
Here, 'ближайшую' describes 'дату' (date), being in the accusative feminine singular to match the noun.
Она выбрала ближайшую дату для встречи.
She chose the nearest date for the meeting.
In this case, 'ближайшую' modifies 'остановку' (stop), again in the accusative feminine singular form as per the noun.