In the previous mishnah the Sages defined who a Nasi is, the one who brings the goat of the Nasi. Now, in mishnah 4 of the third chapter of Maseches Horayos, the Sages come to define who the anointed kohen is - the one who brings the bull of the anointed kohen, that special bull offering.
"Eizehu ha'Mashiach? Ha'mashuach b'shemen ha'mishchah" - Who is the anointed kohen? The one anointed with the anointing oil:
This kohen was anointed with the special anointing oil in order to become Kohen Gadol. Already in Parashas Ki Sisa, the Holy One, Blessed is He, commanded Moshe Rabbeinu to prepare the anointing oil, which was used to inaugurate the Mishkan and its vessels, and to anoint Aharon the Kohen and his sons for the priesthood. Afterward the oil was kept and used to anoint Kohanim Gedolim throughout the entire period of the first Beis HaMikdash.
In the days of King Yoshiyahu, before the destruction of the first Beis HaMikdash, the king hid away the anointing oil so that it would not fall into the hands of the nations. From that time on, there is no way to confer upon a kohen the status of Kohen Gadol through anointing, and the alternative is 'ribuy begadim', the multiplicity of garments: a kohen who is dressed in the garments of the Kohen Gadol is called a "merubeh begadim" - one with the additional garments. The mishnah emphasizes that only the one anointed with the anointing oil is called the anointed kohen, and not the merubeh begadim.
The priestly garments:
An ordinary kohen - wears four garments only: the tunic, the trousers, the turban (a kind of hat) and the sash (a kind of belt).
The Kohen Gadol - receives four additional garments: the eiphod (a kind of apron), the choshen (a kind of breastplate), the me'il (a kind of robe) and the tzitz (a golden headband). It is these four additional garments that grant the kohen the status of Kohen Gadol.
According to the Rambam, merely wearing the garments is enough, while according to Rashi he must actually perform service in the eight priestly garments. In any case, this is the way to become a Kohen Gadol as far as conferring the sanctity of the office, but it does not make him the anointed kohen who is fit to bring the bull of the anointed kohen.
The difference between the anointed kohen and the merubeh begadim:
The mishnah continues, and its words echo the mishnah in Maseches Megillah: "Ein bein kohen ha'mashuach b'shemen ha'mishchah l'merubeh begadim ela par haba al kol ha'mitzvos" - the only difference between a kohen anointed with the anointing oil and one with the additional garments is the bull brought for transgression of any of the mitzvos. This is the case under discussion in Maseches Horayos: a Kohen Gadol who issued a mistaken ruling in a mitzvah for which an ordinary person would be liable to a fixed sin offering for an inadvertent transgression. This is the only scenario and the only difference; in every other respect there is no difference at all between a Kohen Gadol appointed with the anointing oil and the merubeh begadim.
The serving kohen and the kohen who has stepped down:
The mishnah continues, again in the language of the mishnah in Megillah: "V'ein bein kohen ha'meshamesh l'kohen she'avar" - and there is no difference between the serving kohen and the kohen who has stepped down. This matter seemingly departs from the subject of Maseches Horayos, but it is a direct continuation of the discussion of the distinction between the various types of kohanim. The serving kohen is the one actively holding the office, and the kohen who has stepped down is a Kohen Gadol who no longer serves.
For example, at the opening of Maseches Yoma it is described that if the Kohen Gadol became tamei, another kohen is appointed in his place. During the period of his tumah he does not serve, and the other is the Kohen Gadol in office; and once the first becomes tahor, he returns to serve as the functioning Kohen Gadol, and the second is called the kohen who has stepped down, a former Kohen Gadol.
As we learned earlier in the chapter, once the sanctity of a Kohen Gadol rests upon a person, that sanctity remains with him forever. The Mishnah therefore asks: if we have before us two men upon whom the sanctity of a Kohen Gadol has taken effect, one currently serving and one who served previously, and both are alive - what is the halachic difference between them? The answer is "ela par Yom HaKippurim va'asiris ha'eifah" - only the bull of Yom Kippur and the tenth of an eifah.
These two items cannot possibly be performed by two Kohanim Gedolim together. Under ordinary circumstances the Kohen Gadol may serve on a regular day as little or as much as he wishes, while the Yom Kippur service must be performed specifically by him. But the basis of the distinction here is money: the Kohen Gadol is required to pay for these two offerings out of his own pocket, and it is impossible for two Kohanim Gedolim to bear a single payment:
Par Yom HaKippurim - the special bull brought on Yom Kippur to atone for the Kohen Gadol, his household and all the kohanim. The Kohen Gadol pays for it from his personal funds, and therefore the serving kohen is the one who pays for it.
Asiris ha'eifah - the quantity of fine flour used for the 'chavitei Kohen Gadol'. Every day the Kohen Gadol brings a meal offering of a tenth of an eifah, which is 43.2 egg-volumes, and from it he bakes twelve loaves on a griddle. Half are offered in the morning with the daily morning offering and half in the afternoon with the daily afternoon offering (some phrase it as six in the morning and six in the afternoon, and some phrase it as half a dozen in the morning and half a dozen in the afternoon - and there is no difference whatsoever between them).
The Kohen Gadol is not obligated to bring the chavitin himself - any kohen may offer them - but he must pay for them. Only one offering is brought each day, and therefore the payment falls upon the serving kohen and not upon the kohen who stepped down.
In everything else - they are equal:
In all other matters the two stand on common ground, in the words of the Mishnah: "zeh vazeh shavin ba'avodas Yom HaKippurim, um'tzuvin al habesulah, va'asurin al ha'almanah, v'einan mitam'in bikroveihen, v'lo por'in, v'lo por'min, umachzirin es haroze'ach" - this one and that one are equal regarding the Yom Kippur service, and they are commanded regarding the virgin, and forbidden regarding the widow, and they do not become tamei for their relatives, and they do not let their hair grow, and they do not rend their garments, and they cause the killer to return:
"Shavin ba'avodas Yom HaKippurim" - each of them is fit to perform the Yom Kippur service. For example, a Kohen Gadol who stepped down because of old age and then regained his strength, or who had a blemish which then passed - if he performed the service, his service is valid.
"Um'tzuvin al habesulah" - there is a positive commandment for a Kohen Gadol to marry a virgin, and this requirement does not apply to an ordinary kohen. This obligation applies equally to the one serving and to the one who stepped down.
"Va'asurin al ha'almanah" - both are forbidden to marry a widow. An ordinary kohen may marry a widow and is forbidden only to a divorcee, whereas a Kohen Gadol is forbidden even to a widow.
"V'einan mitam'in bikroveihen" - an ordinary kohen is forbidden to contract tumah from a corpse, but he is commanded to become tamei and involve himself in the burial of his seven relatives: his mother and father, his brother and sister (and the sister specifically when she is not married to another man), his son, his daughter and his wife. The Kohen Gadol does not become tamei even for them, not even for his parents, and in this the serving kohen and the one who stepped down are equal.
"V'lo por'in, v'lo por'min" - these are practices of mourning. A mourner does not cut his hair, whereas the Kohen Gadol may not let his hair grow long, but rather must have it trimmed every day. Similarly, rending the garment, which is done in mourning, does not apply to a Kohen Gadol.
"Umachzirin es haroze'ach" - as we learned in Maseches Makkos, one who kills accidentally is exiled to a city of refuge until the death of the Kohen Gadol. The death of either one of them suffices, and even if the Kohen Gadol who stepped down dies, the exile may return home.
Regarding the prohibition of a widow, one must ask: the Kohen Gadol is commanded to marry a virgin, so in what circumstance could marrying a widow even arise for him? The answer lies in the two stages of marriage. The first stage is erusin, also called kiddushin, and legally it constitutes a full marriage. A woman who has been betrothed with a ring - "behold you are betrothed to me with this ring" - is a married woman in every respect, except that she has not yet actually been brought into the marriage and she is still living in her father's house. If her husband dies or divorces her, she becomes a widow or a divorcee while still a virgin. This is the woman who is a virgin and nevertheless forbidden to a Kohen Gadol, because she is a widow.
In summary: in this Mishnah we learned that the anointed kohen is specifically the one anointed with the anointing oil, and not the one inaugurated through the additional garments, and that the only difference between them is the bull brought for all the mitzvos. We also learned that the only difference between the serving kohen and the kohen who stepped down is the bull of Yom Kippur and the tenth of an eifah, for which payment falls upon the serving kohen alone, while in all the other laws of the high priesthood - the Yom Kippur service, the virgin and the widow, tumah for relatives, letting the hair grow and rending the garment, and causing the killer to return - the two are equal.