Bava Kamma, Chapter 7, Mishnah 2. The Mishnah lists a variety of cases where one is indeed liable for the fourfold and fivefold payments, and each of them presents a certain novel point or addresses a fine nuance.
Testimony Regarding the Theft and Testimony Regarding the Slaughter:
The first case deals with testimony that obligates a person to pay. A fundamental rule is that establishing a halachic fact in beis din is done according to the principle of "by the mouth of two witnesses shall a matter be established," and it is understood that a single pair of witnesses must establish the entire halachic fact - we do not combine two pairs of witnesses to piece together the full picture. The question that arises is whether the theft and the slaughter or sale are considered one matter, or two separate matters for which a pair of witnesses for each is sufficient. The answer is the second option: these are two separate halachic matters, and therefore one can bring a pair of witnesses for each of the two stages.
"Ganav al pi shenayim" - It was proven that the thief stole based on the testimony of one pair of witnesses.
"Vetavach umachar al pihem" - Those very same witnesses testify that the thief also slaughtered or sold the animal he stole.
"O al pi shenayim acherim" - Alternatively, one set of witnesses confirmed that he stole, and a second set confirmed that he slaughtered or sold.
"Meshalem tashlumei arbaah vachamishah" - In both cases the thief is liable for the fourfold or fivefold payments, since these are two separate matters and two pairs of witnesses can establish them.
Cases Involving an Accompanying Prohibition:
"Ganav umachar beShabbos" - One who steals an animal and sells it to another on Shabbos is liable for the fourfold and fivefold payments. The text uses the term "sold" and not "slaughtered," because if he had slaughtered the animal on Shabbos, he would not pay four and five: by slaughtering on Shabbos he violates a Torah prohibition, and we enter the rule of "kim lei bederabbah mineih," which we have already seen earlier in the tractate. This rule applies when the case involves a death penalty, such as desecrating Shabbos, and even if the person is not actually executed by beis din - since this is a transgression for which one incurs the death penalty, we do not impose a monetary punishment upon him as well. As we will see in Mishnah 4, it is possible that he will be liable for the first stage, the double payment, since the sale itself does not involve a Torah desecration of Shabbos; but for the second stage, if he had slaughtered the animal, he will not pay the three additional payments because of kim lei bederabbah mineih. While it is true that it is forbidden to sell things on Shabbos, this is only a rabbinic law and does not trigger the rule of kim lei bederabbah mineih.
"Ganav umachar la'avodah zarah" - One who steals an animal and sells it to a Jew who is about to use it as an offering for idolatry is liable. Although it is forbidden to sell animals to those who will offer them for idolatry, this is merely the prohibition of "lifnei iver" (placing a stumbling block before the blind) - a Torah prohibition that does not carry a death penalty, and therefore he is liable for the fourfold and fivefold payments. But if he himself had slaughtered the animal for idolatry, this would be a transgression carrying a death penalty, like the law for all idolatry, and he would be exempt from the fourfold and fivefold payments.
"Ganav vetavach beYom HaKippurim" - One who stole and subsequently slaughtered, and certainly if he sold on Yom Kippur, is liable. The reason is that even though Yom Kippur is called "Shabbos Shabbaton," regarding the punishment for performing melachah on it, the halachah is that one who performs melachah without witnesses and forewarning receives the punishment of kares, while with witnesses and forewarning his punishment is lashes. One who receives "only" kares is not considered to be transgressing a sin carrying a death penalty according to our Tanna, and therefore he is liable for the fourfold and fivefold payments. However, if there were witnesses who warned him: "Do not slaughter the animal on Yom Kippur, for if you do you will receive lashes," and he replied that he does not care and is doing it anyway - they would administer lashes to him; and it is a rule that once a person receives lashes, he is exempt from monetary payments. It turns out that had there been a warning and he would receive lashes, he would be exempt from the fourfold and fivefold payments as well.
Stealing from His Father and Consecrating after Slaughtering:
"Ganav mishel aviv vetavach umachar ve'achar kach mes aviv" - One who steals from his father, slaughters the animal or sells it, and only afterward his father dies, is liable to pay four and five. There is no great novelty here: he pays four and five to the estate, and even if he subsequently inherits part of this money, so what. The novelty will become clear in Mishnah 4, in the case where the father dies before the sale or slaughter.
"Ganav vetavach ve'achar kach hikdish" - He stole an animal, slaughtered it, and subsequently consecrated the meat of the dead animal to the Holy Temple - "meshalem tashlumei arbaah vachamishah". The reason is that at the time he stole and at the time he slaughtered, the animal was under private ownership, and property under private ownership is what obligates one in the fourfold and fivefold payments, as we will see in Mishnah 4. From the moment it transferred to the ownership of the Holy Temple as hekdesh, he is exempt from the payments, and we will expand on the basis for this in Mishnah 4.
The Definition of "Slaughtering" - Valid Shechitah:
In the next part of the Mishnah, it is explained that the reference to one who "slaughters" means specifically a valid shechitah. The question is what the law is when he slaughtered in a manner that constitutes a valid shechitah, but he did not intend to eat the animal. One might have thought that there is no liability for four and five, but this is not so: he is still liable, for a shechitah is a shechitah.
"Ganav vetavach... lirefuah" - One who slaughters for a medical purpose, such as to extract blood from the animal in order to soak his feet in it and the like, and his goal is not to eat the animal.
"O likelavim" - He slaughtered it not for his personal needs nor for human consumption, but rather to feed it to his dogs. Even though it will not be eaten by humans, he is liable.
"Hashocheit venimtzeit tereifah" - Even if he slaughtered it and it turned out that it was halachically unfit for human consumption because it was a tereifah, meaning it had a defect indicating it would die within twelve months, such as a hole in the heart or lungs. No one can eat it, and yet he is still liable for the fourfold or fivefold payments.
"Hashocheit chullin ba'azarah" - One who slaughters the animal in the azarah, the Temple courtyard, does something strictly forbidden, and chullin (non-sacred animals) slaughtered in the azarah are forbidden to be eaten. Even though they will not be eaten, he is liable for the fourfold and fivefold payments.
Regarding the last point, there is a technical side note: chullin slaughtered in the azarah are forbidden for any benefit, and since this is the case, the animal is not halachically considered his. Therefore, he should have been exempt from the fourfold and fivefold payments for a technical reason. Because of this, the Mishnah must be referring to a situation where the slaughter began outside the azarah and was completed inside it - in this way, the animal is considered his to an extent that makes him liable for slaughtering it, even though it is forbidden to be eaten because it is considered an animal slaughtered in the azarah.
The opinion of Rabbi Shimon:
Rabbi Shimon disagrees regarding the last two points. He also agrees with the principle that a kosher slaughter is required, but they argue about its definition. The Tanna Kamma holds that the definition of slaughter is technical: cutting the majority of the two simanim, the esophagus and the windpipe, and performing this technical action is sufficient. Rabbi Shimon, on the other hand, holds that slaughter is essentially defined by its purpose: slaughter is a permit, allowing the meat of the animal to be eaten. Slaughter turns an animal that cannot be eaten, like a tereifah or a neveilah (essentially a neveilah waiting to happen), into a kosher animal fit for consumption. Therefore, if the act of slaughter does not permit the animal to be eaten, it is not considered slaughter but merely a generic killing.
Therefore, "Rabbi Shimon poter bishnei eilu" - in these last two cases, a tereifah and chullin in the azarah, he is exempt from the fourfold and fivefold payments, since in both of them, even if he slaughtered properly, the animal cannot be eaten, either because it is a tereifah or because it was slaughtered in the azarah. This is not called slaughter but rather a mere technical act of slaughtering, and it does not meet the definition that would make him liable for four or five.
Halachah: The halachah does not follow Rabbi Shimon but rather the Tanna Kamma, and therefore, in all the cases listed by the Mishnah, the person will be liable for the fourfold and fivefold payments.