This is the 9049th most frequent Polish word.
starej
"Starej" means "of the old" or "old (feminine, genitive case)" in English.
In this sentence, 'starej' is the feminine singular form of the adjective 'stary', meaning 'old', in the locative case used to describe the noun 'biblioteka'.
Idę do starej biblioteki.
I go to the old library.
Here, 'starej' describes the feminine singular noun 'kobieta', indicating the adjective 'stary', meaning 'old', in the dative case.
Pomagam starej kobiecie.
I help the old woman.
In this example, 'starej' modifies the feminine singular noun 'kamienica', expressing the locative case of the adjective 'stary', meaning 'old'.