This Mishnah, Bechoros chapter 2, Mishnah 8, is the last in a series of three Mishnayos dealing with a doubtful bechor, and in fact repeats and teaches the same principle for the third time. The case: two sheep in a pen - one is giving birth for the first time, and the second is not giving birth for the first time. Only the offspring of the one giving birth for the first time can be a bechor, while the offspring of the one not giving birth for the first time is not a bechor at all, whether it is a male or a female.
The First Case - Two Males:
The owner finds two male offspring in the pen. He does not know which offspring belongs to which mother, but it is clear that the offspring born to the one giving birth for the first time - whichever one it is - is a bechor. Therefore, one of the males will remain in the hands of the owner and one will be given to the kohen, and the question is which one.
In the opinion of Rabbi Tarfon, the kohen takes the better of the two, and this is a certainty. The logic is that the bechor is more robust and stronger than the offspring that come after it, and therefore if one of them is superior and more robust than the other, we can safely assume that it is the bechor. (At this stage we still maintain that Rabbi Tarfon views this as a definite reality, and at the end of the Mishnah, as in the previous two, he retracts this assumption and abandons it.)
Rabbi Akiva disagrees and holds that we cannot assume this, and it is a doubt. Nevertheless, since there are two animals here and one of them certainly goes to the kohen, the dispute is only about the difference between them: if one is worth three and the second four, then three certainly go to the kohen, and the entire debate revolves around the fourth - around the extra fat in the more robust animal.
Therefore, the kohen takes the lesser of the two and holds it in his possession as a doubtful bechor: he must wait until it develops a permanent blemish, and only then may he eat it. The second remains in the owner's possession and he leaves it to graze until it develops a permanent blemish, and at that time he slaughters it and eats it. Until then he is not permitted to do anything with it - not to work with it and not to shear it.
The Obligation of the Gifts for the Animal Remaining in the Hands of the Owner:
When the owner eventually slaughters the doubtful bechor in his possession, seemingly he must give the gifts to the kohen. The logic is that no matter how the matter is decided, the gifts belong to the kohen: if it is a non-sacred animal - in all non-sacred animals the gifts are given to the kohen; and if it is a bechor - the entire animal belongs to the kohen.
"Rabbi Yosi poter" - Rabbi Yosi exempts, following his approach in the two previous Mishnayos. In his opinion, we relate to the division between the owner and the kohen as a complete barter transaction: the kohen received his portion, and in return he transferred all his rights and relinquished all his claims to the second animal, such that it belongs entirely to the owner, and there are no gifts in it at all. As he states: "Kol shechalifav beyad kohen, patur min hamatanos" - Anything whose exchange is in the hand of a kohen is exempt from the gifts - and here the rule is stated in full.
"Rabbi Meir mechayev" - Rabbi Meir obligates, meaning there is an obligation to give the gifts. This is the first time Rabbi Meir's name is mentioned here, even though it is the same approach taught previously, and from now on it is known that Rabbi Meir is the one who obligates the gifts. The halachah in this case follows Rabbi Meir.
"Mes echad meihen" - If one of them died:
The two mothers are as described above, and in the morning two males were found in the pen, except that one of them died before the kohen received his portion. One offspring remains and it is not known who its mother is: if it is the offspring of the one giving birth for the first time - then it is a bechor; and if it is the offspring of the one not giving birth for the first time - it is not a bechor.
"Rabbi Tarfon omer: Yachaloku" - Rabbi Tarfon says: They should divide it - we evaluate the offspring and divide its value between the owner and the kohen, half and half. Seemingly this is puzzling: didn't Rabbi Tarfon hold that the stronger of the two certainly belongs to the kohen? Rather, here too, for the third time, he retracts his assumption and holds that it is a doubt (historically he retracted only once, but the Mishnah teaches it three times).
And if it is a doubt, we must ask why we do not apply the rule of "the burden of proof is on the one who seeks to extract from his friend." This is indeed the argument of Rabbi Akiva: "Hamotzi mechavero alav hara'ayah" - the kohen cannot prove that anything is owed to him, and therefore he takes nothing, so why should they divide it half and half?
The answer, as in the previous Mishnah, is a specific establishment of the case: the Gemara establishes the case as one where the owner was a Yisrael who hired a kohen as a shepherd and transferred certain rights to him, and for this reason they divide it between them. Rabbi Akiva rejects this establishment and holds that people do not typically act in this manner.
"Zachar unekevah" - a male and a female:
This is the fourth and final combination, completing all the possibilities. Two mother sheep - one a first-time mother and the other not a first-time mother - and in the morning, two offspring were found in the pen: a male and a female. The best situation from the kohen's perspective is that the first-time mother gave birth to the male, and then there is a firstborn here; but it is possible that the first-time mother gave birth to the female, and the male came from the mother that is not a first-time mother. Therefore, the kohen cannot prove that he is entitled to anything, it is a doubt, and the kohen gets absolutely nothing here.
The male remains in the possession of the owner, and since it might be a firstborn, it is restricted by all the prohibitions that apply to a firstborn: it is forbidden to shear it or work with it. The owner leaves it to graze, and when it develops a permanent blemish, he slaughters it and eats it, and the halachah is that he must give the gifts to the kohen.
It is worth noting that the ruling that he gives the gifts to the kohen is actually subject to a dispute between the Rambam and the Tur, and therefore we must be reserved about making an absolute determination: according to the Rambam, it is possible that the gifts remain in the possession of the owner, while the Tur holds that they are given to the kohen.
In summary: In this Mishnah, the combinations of a doubtful firstborn with two mothers, a first-time mother and one that is not a first-time mother, were taught: in the case of two males - Rabbi Tarfon awards the kohen the better of the two, and Rabbi Akiva sees this as a doubt and the dispute is only over the difference between them; if one of them died - Rabbi Tarfon holds "they divide it" (and this is established as a case of a kohen shepherd), and Rabbi Akiva adopts the principle of "the burden of proof is on the one who seeks to extract from his friend"; and in the case of a male and a female - the kohen gets nothing, and the offspring grazes in the owner's possession until it develops a blemish. Regarding the gifts, Rabbi Yosi exempts, based on the law of exchange, and Rabbi Meir obligates, and the halachah follows Rabbi Meir.