The firstborn must be eaten “year by year”. What does this mean? If a firstborn was born blemished, it does not have the full holiness of the firstborn and therefore is allowed to be eaten. In that case it must be slaughtered and eaten within a year from its birth.
If the firstborn was born unblemished then in the time of the Temple it becomes eligible to be offered in the Temple after eight days. Within a year after this time it must indeed be brought as a sacrifice, slaughtered, and eaten by a Kohen and his family. Today an unblemished firstborn poses a problem: the owner must care for it and then give it to a Kohen. He cannot force the kohen to accept it, because it is a present, so if he cannot find a Kohen, he continues to care for it until it develops a blemish, however long this takes.
One can only slaughter a firstborn through the permission of an expert. If one first slaughtered it, and then showed it to an expert, Rabbi Yehudah permits it to be eaten, but Rabbi Meir forbids it because of a possible confusion.
Art: Lovis (Franz Heinrich Louis) Corinth - In The Slaughter House