This is the 3991st most frequent Russian word.
калининградской
"Kaliningradskoy" refers to something related to Kaliningrad, a city and region in Russia.
This usage of "Калининградской" is an adjective in the prepositional case, describing the library (feminine noun) being located in Kaliningrad.
Я купил книгу в Калининградской библиотеке.
I bought a book in the Kaliningrad library.
Here, "Калининградскую" is an adjective in the accusative case, modifying "область" (region) and describing it as related to Kaliningrad.
Мы посетили Калининградскую область во время отпуска.
We visited the Kaliningrad region during our vacation.
In this sentence, "Калининградской" is an adjective in the genitive case, associated with "области" (of the region) and indicating possession or relation.