Jake was a black crow (grey pigeon in the book) owned by Brooks Hatlen. (Sherwood Bolton in the book)
Brooks, the Shawshank prison librarian, found Jake as an infant that fell out of his nest outside the plate shop and injured its wing. He took the crow in and nursed it back to health, naming him "Jake" and keeping him as a pet, feeding him maggots he finds in the prison food. When Brooks was granted parole and got out of jail, he knew that he had to free Jake, probably his closest friend, and did so with great hesitation and pain. Afterwards Brooks writes about how much he misses him and sometimes hoping he'll appear with the birds he sees out there.
Jake was a pigeon in the book and was owned by Sherwood Bolton instead of Brooks.