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 as Jake as a hatchling that fell out of his nest outside the plate shop with an injured wing. So, he took the crow in and nursed it back to health, naming him "Jake" and keeping it as a pet, feeding him maggots he finds in the prison food. When Brooks was paroled 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 tells the audience about how much he misses him and sometimes thinks of him as being one of the birds he sees out there.
Jake was a pigeon in the book and was owned by Sherwood Bolton instead of Brooks.