This is the 7769th most frequent Russian word.
ильинском
"Iльинском" is the prepositional or locative case of "Ильинский," meaning "related to Ilyinsky" (a place or surname in Russia).
The word "Ильинскому" refers to the name "Ильинский" in the dative case, which is used in this sentence to describe the location (dative used with 'по').
Мы отправились в поход по Ильинскому лесу.
We went on a hike through the Ilyinsky forest.
The word "Ильинскому" is in the dative case here again, describing the dative subject of the sentence referring to a monastery's characteristic.
Ильинскому монастырю уже больше пяти веков.
The Ilyinsky monastery is over five centuries old.
Here, "Ильинскому" in the dative case modifies the specific location described in the sentence (indicating movement or relating to).