We just learned that although one can redeem a firstborn donkey with any item of equal value, there is a special law permitting its redemption with a seh - a goat or a sheep - even if its value is less than that of the donkey, thereby permitting one to benefit from the donkey. Our Mishnah comes to list which animals are invalid for this redemption.
These are the ones with which one may not redeem:
"Ein podin lo be'egel" - One may not redeem with a calf whose value is less than the firstborn donkey, since it is not a sheep or goat.
"Velo bechayah" - One may not redeem with an undomesticated animal, such as a deer, which is not included in the category of a sheep or goat (in a case where its value is less than the firstborn donkey).
"Velo besh'chutah" - One may not redeem with a goat or sheep that was slaughtered, when the value of the slaughtered animal is less than the value of the firstborn donkey. The reason: A gezeirah shavah (verbal analogy) connects this law to the sheep or goat of the Pesach offering - just as one does not bring a slaughtered animal for the Pesach offering, so too one may not redeem the firstborn donkey with it.
"Velo betereifah" - A tereifah is an animal that has sustained a fatal defect and cannot survive. For example, if the leg of the goat or sheep was cut off above the knee in a way that the animal cannot survive, or if it is clearly known to have another fatal defect, such as a hole in the heart. Here too, the law is derived through a gezeirah shavah from the fact that one does not bring such an animal for the Pesach offering, which is also a sheep or goat.
"Velo bekilayim" - Kilayim is a crossbreed: A male goat that mated with a female sheep, and the offspring born from them. This reality indeed exists, although it is rare and occurs approximately once a decade. According to the Sages, such an offspring is not considered a sheep or goat, as it is a sort of mixture - a half-breed.
"Velo bekoy" - The koy is a species concerning which there is a doubt whether it is a domesticated animal or a wild beast, similar to a wild goat. (As an example, one could point to the bezoar ibex, an ancestor of domesticated goats, which looks like a goat but has a wild appearance.) The exact identity of the koy is unknown, but in any case, one may not redeem with it.
The opinion of Rabbi Elazar:
Rabbi Elazar "matir bekilayim" - permits a crossbreed. For technical reasons that are beyond our current scope, he holds that an offspring born from a male goat and a female sheep is also considered a sheep or goat, "mipnei shehu seh" - because it is a sheep or goat. However, "ve'oser bekoy" - he prohibits a koy. He concedes that one may not redeem with a koy because its status is doubtful: perhaps a koy is a sheep or goat, and perhaps it is not, and due to this doubt, it may not be used for redemption.
"Netano lekohen":
The Mishnah does not explain what this refers to, but rather introduces a new topic: an Israelite who gave the unredeemed donkey to a kohen, telling him that he does not have a sheep or goat in his possession and wishes to rid himself of the donkey. In this case, the Mishnah rules, "ein kohen rashai lekayemo ad sheyafrish seh tachtav" - the kohen is not permitted to keep the donkey in his possession until he redeems it with a sheep or goat.
The basis of the law is a Rabbinic enactment: The Israelite who gives his donkey to a kohen must see the kohen perform the redemption on the spot. The reason for this concern is that the kohen might say to himself: Why should I bother with the redemption? After all, it makes no practical difference, since the donkey is permitted for me to benefit from after the redemption, and the sheep or goat with which I redeem it also remains in my possession and is permitted to me, so there is no financial difference whether I redeem it or not. Due to this concern that they might be lazy and not bother to redeem it, it would result in the mitzvah of the Torah not being fulfilled.
Therefore, an Israelite who was lazy and gave the unredeemed donkey to a kohen must stand there and watch the kohen take an item equal in value to the donkey, or a sheep or goat, on the spot, and redeem the donkey with it. Afterward, the kohen is permitted to benefit without any restriction from both the donkey and the item with which he redeemed it.
In summary: In this Mishnah we learned the list of animals with which one may not redeem a firstborn donkey - a calf, a wild beast, a slaughtered animal, a tereifah, a crossbreed, and a koy - and the source for these laws from a gezeirah shavah to the sheep or goat of the Pesach offering; the opinion of Rabbi Elazar who permits a crossbreed because it is a sheep or goat, and prohibits a koy due to doubt; and the Rabbinic enactment that an Israelite who gives his unredeemed donkey to a kohen must see the redemption actually being performed, lest the kohen be lazy and the mitzvah be neglected.