Chagigah, Chapter Two, Mishnah 4. The dispute recorded in the previous Mishnah between Beis Shammai and Beis Hillel - whether the olas re'iyah (the pilgrimage burnt offering) may be brought on Yom Tov or not - is what generates the dispute before us.
"Atzeres shechal lihyos b'erev Shabbos":
Shavuos is only a single day, and here the Mishnah is discussing a case in which it falls on Friday. Beis Shammai and Beis Hillel disagree about when the pilgrimage offerings are to be brought:
Beis Shammai say: "Yom tevoach achar haShabbos" - the day of offering the sacrifices takes place on the Sunday after Shabbos. This is because in their view one does not bring an olas re'iyah on Yom Tov, but only shalmei chagigah (the festival peace offering). There is therefore no way to bring it on Shavuos that falls on Friday, and certainly not on the Shabbos that follows. The first opportunity to bring it is on Sunday, and that day is called "yom tevoach" - the day of slaughtering and bringing the offerings.
Beis Hillel say: "Ein yom tevoach achar haShabbos" - there is no need for it, since in their view one may bring the olas re'iyah on Yom Tov itself, that is, on Shavuos that falls on Friday, and there is no reason to wait until Sunday.
However, "Umodim she'im chal lihyos b'Shabbos sheyom tevoach achar haShabbos" - if Shavuos falls on Shabbos, even Beis Hillel agree that the day of offering will be Sunday, and not only regarding the olas re'iyah but even regarding the shalmei chagigah. The reason is that bringing the olas re'iyah and the shalmei chagigah overrides Yom Tov but does not override Shabbos, and since the Yom Tov itself is Shabbos, bringing them is pushed off to the following day.
The dispute with the Tzedukim in the background:
In every case where the offering is pushed off to Sunday - whether according to Beis Shammai when Shavuos falls on Friday, or according to both Beis Shammai and Beis Hillel when Shavuos falls on Shabbos - Chazal had a particular interest in seeing to it that this Sunday should not appear overly festive. The reason lies in the position of the Tzedukim (Sadducees), who maintained that Shavuos must always fall on a Sunday: since they did not believe in the Oral Torah, they read the verse literally and interpreted "mimochoras haShabbos hashevi'is" as referring to an actual Shabbos, and therefore Shavuos, which follows it, always falls on a Sunday.
We, however, know that this is not the meaning of the verse: the day of bringing the Omer, which is the first "mimochoras haShabbos" mentioned in the Torah, alludes to the fifteenth of Nissan, the first day of Pesach, and the day after it is "mimochoras haShabbos." On that day the Omer is brought, and seven weeks later Shavuos falls.
Playing down the festive character of that day:
For this reason, when the offerings are brought on the Sunday adjacent to Shavuos, we do not want it to look as though we agree with the Tzedukim and are lending strength to their position. The Mishnah therefore says "Ein kohen gadol mislabesh b'keilav" - the Kohen Gadol does not dress in his garments, and there are two ways to explain this:
Outside the Beis HaMikdash: on every Yom Tov the Kohen Gadol would wear especially festive and distinguished clothing even when he was not performing the service, and on this day he does not wear them, so that the day should not appear too festive.
During the service: he does not wear the eight garments of the Kohen Gadol, which he is required to wear when he comes to perform the service. In other words, the Kohen Gadol makes a point of not performing the service on that day, in order to play down its festive character.
Similarly, although ordinarily one who brings an offering is not permitted to deliver a eulogy or to fast, on this day he is permitted to do so - all in order not to give effect to the claim of those Tzedukim who say that Atzeres always falls after Shabbos, for we do not want to give their words even the appearance of support.