Mishnah 1 of chapter 2 of Maseches Temurah follows directly from the previous mishnah. There we learned that an individual's offering - an offering a private person purchased with his own money - can be subject to the laws of temurah, whereas a communal offering, purchased with public funds for the Beis HaMikdash, is not subject to temurah. Building on that, our mishnah goes on to list additional general differences between communal offerings and individual offerings.
The mishnah opens: "Yesh b'korbanos hayachid mah she'ein b'korbanos hatzibbur, v'yesh b'korbanos hatzibbur mah she'ein b'korbanos hayachid" - there are laws that apply specifically to the offering of a private person and do not apply to offerings purchased with public funds, and there are laws that apply specifically to communal offerings and do not apply to an individual's offering.
The three laws unique to individual offerings:
"Korbanos hayachid osin temurah v'korbanos hatzibbur einan osin temurah" - with a private person's offering one can make a temurah, that is, consecrate a second animal in its place, whereas with a communal offering one does not make a temurah.
"Korbanos hayachid nohagin b'zecharim uvinkeivos" - with an individual offering there is the option of bringing either a male animal or a female animal. In practice this refers to a shelamim offering, which may come from either males or females. Communal offerings, by contrast, "einan nohagin ela b'zecharim" - all offerings brought in the name of the community are males only, and even the shelamim brought in the name of the community on Shavuos come from males.
"Korbanos hayachid chayavin b'achrayusan uv'achrayus niskeihen" - even when a set time has been fixed for an individual's offering, a person is not exempt from his obligation once that time has passed; rather, he remains responsible for it and brings it afterward.
For example, a woman after childbirth, who is meant to bring her offering on the fortieth or the eightieth day after giving birth (depending on whether she gave birth to a boy or a girl), and likewise a metzora, who must bring his offerings on the eighth day of his purification process: if they missed the day fixed for bringing the offering, they are not exempt from their obligation, but bring it at a later time.
The same applies to responsibility for the nesachim. In general, olos and shelamim require nesachim - a wine libation that accompanies the offering, as well as a meal offering of nesachim that comes with oil. If a person did not bring the nesachim that accompany his offering at their proper time, he remains responsible for them and brings them afterward.
By contrast, "korbanos hatzibbur ein chayavin lo b'achrayusan v'lo b'achrayus niskeihen" - with a communal offering, if the opportunity passed and the offering was not brought at its proper time, there is no longer any place to bring it after the fact, and the same applies to bringing the nesachim.
However, the mishnah adds: "Aval chayavin b'achrayus niskeihen mishekarav hazevach" - if the offering itself was brought properly, but the nesachim were not brought for whatever reason, the nesachim may be brought at a later time, even many days afterward.
The two laws unique to communal offerings:
The mishnah states: "Yesh b'korbanos hatzibbur mah she'ein b'korbanos hayachid, shekorbanos hatzibbur dochin es haShabbos v'es hatumah":
Overriding Shabbos: an offering that the community is obligated to bring is offered even on Shabbos.
Overriding tumah: if all the kohanim are tamei, the service is performed in a state of tumah. Ordinarily, when the kohanim are tamei the offering is invalid, but with a communal offering it is brought nonetheless, and even though the service was performed in a state of tumah, the offering is valid.
Whereas "korbanos hayachid einan dochin lo es haShabbos v'lo es hatumah" - a private person's offering does not override Shabbos. A woman after childbirth whose day for bringing her offering falls on Shabbos does not bring it on Shabbos, and likewise, if all the kohanim are tamei, one must wait until they become tahor, and only then is the individual's offering brought.
The view of Rabbi Meir:
Rabbi Meir disagrees with this definition and holds that whether an offering overrides Shabbos and tumah does not depend on whether it is a communal offering or an individual's offering. In his words: "Vahalo chavitei kohen gadol ufar Yom HaKippurim korban yachid, v'dochin es haShabbos v'es hatumah" - Are not the griddle-cakes of the Kohen Gadol and the bull of Yom Kippur individual offerings, and yet they override Shabbos and tumah? He lists offerings that are purchased by a private person with his own money, and nevertheless override Shabbos and tumah:
The griddle-cakes of the Kohen Gadol: the daily chavitin meal offering that the Kohen Gadol brings and pays for out of his own pocket, half in the morning and half in the afternoon. Halachically this is an individual's offering, since it is not purchased with communal funds, and even so it is brought every single day, including Shabbos, and it also overrides tumah.
The bull of Yom Kippur: the Kohen Gadol is obligated to buy a bull with his own money and bring it in the Beis HaMikdash on Yom Kippur, and its blood is brought into the Holy of Holies (along with the goat offered to Hashem, as opposed to the goat sent away to Azazel). This bull atones for the Kohen Gadol, for his household and for the kohanim. Since the Kohen Gadol pays for it privately, it is an individual's offering, and as we learn from the verses, even if Yom Kippur falls on Shabbos the Kohen Gadol certainly brings the bull.
The Pesach offering: every person's Pesach offering is an individual's offering, and nevertheless, if Pesach falls on Shabbos, it overrides Shabbos and tumah.
Rabbi Meir's conclusion: "Ela shezmanan kavua" - the question is not whether we are dealing with an individual's offering or a communal offering, but whether the offering has a fixed time on the calendar, so that it is not valid at any other time.
Accordingly, there are communal offerings that have no fixed time, and therefore do not override Shabbos and tumah: the bull brought for a mistaken ruling of the community, and likewise the goat brought for idolatry. Although both are purchased with communal funds, they have no set date, and therefore their offering is postponed until after Shabbos.
The Bartenura rules that the halachah follows Rabbi Meir, and indeed that is the case. The Tosafos Yom Tov, however, points out that in practice there is no great dispute between them: Rabbi Meir states the precise law, while the Tanna Kamma is stating a broad rule - a communal offering overrides Shabbos and tumah and an individual's offering does not - and the offerings that Rabbi Meir listed are simply exceptions to the rule. According to the Tosafos Yom Tov there is therefore no real disagreement here, and in his opinion the Bartenura is not precise in his wording when he writes that the halachah follows Rabbi Meir, as though the Tanna Kamma disputes these laws - which he apparently does not.