The mishnah before us is the last in chapter 3 of Maseches Eduyos, and it closes the list of Tannaim recorded in it in descending order of age. The last of them is Rabbi Elazar ben Azaryah, who at that time was eighteen years old according to the Babylonian Talmud, and sixteen according to the Jerusalem Talmud. He too has three additional halachos in which he disagreed with the Sages: he was lenient and permitted, while the Sages forbade.
"Paraso yotze'ah birtzu'ah shebein karneha":
Taken at face value, this speaks of a cow that went out on Shabbos with a strap tied between its horns. The Gemara points out that Rabbi Elazar ben Azaryah was wealthy and owned many cows, so it would make no sense to attribute a single cow to him; rather, in fact it was his neighbor's cow, and she was the one who allowed her cow to go out with a strap. Rabbi Elazar ben Azaryah did not protest against her, and for that reason the mishnah places the responsibility on him and says "his cow."
The basis of the law: the Torah commands us "so that your ox and your donkey may rest" - a person is obligated not to cause his animal to perform labor on Shabbos, any of the thirty-nine melachos, including carrying and transferring an object in a public domain. And just as a person himself is forbidden to transfer objects in a public domain and nevertheless may go out wearing his clothing and his ornaments, since they are subordinate to his body and are not considered a separate burden upon him - so too with an animal.
Therefore, items intended to protect the animal, to keep it warm, and even to adorn it - provided these are ordinary, accepted ornaments - are permitted. The question, then, concerns the strap between the cow's horns: is it a normal practice to adorn a cow with it, or not?
Rabbi Elazar ben Azaryah was lenient and held that the strap is for beauty and adornment, and is not considered a burden upon the animal, and therefore it may go out with it on Shabbos. The Sages disagreed and ruled that this is not a normal way to adorn animals, and the strap is not in the category of an ornament but an actual burden, and therefore one may not allow the cow to go out with it on Shabbos.
"U'mekardin es habeheimah b'Yom Tov":
Rabbi Elazar ben Azaryah permitted combing an animal with a dense metal comb, one with small sharp teeth, in order to remove ticks and lice from it - the way one combs a child to remove lice from his hair. The concern is that scratching with the metal comb is liable to cause the animal to bleed, and causing bleeding on Shabbos or Yom Tov is a melachah (in the category of slaughtering).
Rabbi Elazar ben Azaryah's argument is that this will not necessarily happen, and even if it does happen - the person does not intend it and has no interest whatsoever in his animal bleeding. It emerges that the matter depends on the dispute that runs through all the sugyos: Rabbi Yehudah forbids an unintended act (davar she'eino miskavein), and Rabbi Shimon permits it, and the practical halachah follows Rabbi Shimon. This is the approach Rabbi Elazar ben Azaryah took in our mishnah: even if bleeding results, he did not intend it and he is not performing the melachah. Rabbi Yehudah, by contrast, forbids even combing with a metal comb - and so too is the halachah, as will be explained below.
Grinding pepper on Yom Tov:
The third matter in which he disagreed: Rabbi Elazar ben Azaryah permitted grinding pepper on Yom Tov in a small mill designated for that purpose, and the Sages did not permit it. The concern here is because of uvdin d'chol - activities that are normally done on weekdays. The precise definition of this prohibition is difficult to pin down, and it seems to be subject to a certain degree of assessment: the Sages relied on practical recognition, that when one sees such an activity one knows how to identify it.
By way of conjecture alone one may add: in the time of the Mishnah many things were ground, and grinding - first and foremost the grinding of flour - was a grueling and exhausting labor, one of the chores everyone sought to avoid. Because grinding in a small mill resembles it, it looks like a weekday activity. Either way, the Sages held that grinding pepper, even in a small mill in which one does not prepare a large quantity, is forbidden, and so is the practical halachah. The Mishnah Berurah writes that even such grinding is not permitted on Shabbos.
The view of Rabbi Yehudah and the Sages:
The Mishnah has not yet concluded. Rabbi Yehudah, who holds that an unintended action is forbidden, says: "ein mekardin es habeheimah b'Yom Tov" - one may not curry an animal on Yom Tov with the sharp metal comb whose teeth are set close together, and the same applies on Shabbos, because it makes a wound and is liable to cause bleeding, and this prohibition applies even when the action is unintended. "Aval mekartzfin" - one may use a wooden comb whose teeth are further apart, since it is not liable to cause the animal to bleed.
The Sages say: "ein mekardin v'af lo mekartzfin" - neither with a wooden comb nor with a metal comb, a decree lest one use wooden combs today and metal combs, which are forbidden, tomorrow. And the halachah follows the Sages: one may not comb an animal on Shabbos with any implement at all. The concern is that the combing will pull out hairs; and although we hold that an unintended action is permitted, that does not apply in a case of 'pesik reisha' - where the melachah will certainly be performed. Pulling out the animal's hairs, and likewise causing it to bleed, fall into the category of pesik reisha, since it is virtually certain that they will occur.
In summary: all three of Rabbi Elazar ben Azaryah's leniencies were rejected as halachah, and in each case the ruling follows the Sages: a cow may not go out with the strap between its horns, one may neither curry nor comb an animal on Shabbos or Yom Tov, and one may not grind peppercorns on Yom Tov.