Before us is Mishnah 6 in the third chapter of Maseches Moed Katan, which deals with the question of which festivals interrupt the seven days of mourning and which do not. The starting point is the basic difference between Shabbos and Yom Tov: Shabbos does not interrupt the shivah, whereas Yom Tov does interrupt it.
The view of Rabbi Eliezer:
"Rabbi Eliezer omer: mishecharav Beis HaMikdash, Atzeres k'Shabbos" - after the Beis HaMikdash was destroyed, the law of Shavuos with regard to interrupting the shivah is like that of Shabbos, which does not interrupt the counting of the seven days.
The reason for the difference between the time of the Beis HaMikdash and the time after the destruction: when the Beis HaMikdash stood, Shavuos was extended in a certain sense to seven days, since one who did not bring on Yom Tov the offerings he was obligated in - the olas re'iyah and the chagigah - was able to bring them during the six days following the festival. In this respect Shavuos was a full seven day festival, and therefore its law was like that of the other festivals and it interrupted the shivah. Once the Beis HaMikdash was destroyed this distinction lapsed, and in Rabbi Eliezer's opinion Shavuos no longer interrupts the shivah.
Rabbi Eliezer adds: "Rosh Hashanah v'Yom HaKippurim k'regalim" - the law of Rosh Hashanah and Yom Kippur is like that of one of the three festivals, Pesach, Shavuos and Sukkos, which interrupt the shivah. These days are not counted at all toward the seven days, as we mentioned earlier.
The view of the Chachamim:
"VaChachamim omrim: lo k'divrei zeh v'lo k'divrei zeh" - they do not hold like Rabbi Eliezer, neither regarding the law of Shavuos nor regarding the law of Rosh Hashanah. In their opinion:
"Atzeres k'regalim" - Shavuos has the status of a festival even after the destruction of the Beis HaMikdash, and therefore it interrupts the shivah.
"Rosh Hashanah v'Yom HaKippurim k'Shabbos" - they do not have the status of a festival, and they do not interrupt the shivah.
The halachah:
The halachah does not follow Rabbi Eliezer regarding the law of Shavuos: we rule that Shavuos interrupts the shivah like any other Yom Tov.
The halachah does not follow the Chachamim regarding the law of Rosh Hashanah and Yom Kippur: we rule that their status is like that of a festival with regard to shivah, and they interrupt the shivah. Therefore, one who was sitting shivah before Rosh Hashanah or before Yom Kippur - the arrival of the day brings his mourning to an end.
In summary: in this Mishnah the Tannaim disagree about the status of Shavuos, Rosh Hashanah and Yom Kippur with regard to interrupting the shivah. Rabbi Eliezer held that after the destruction Shavuos is like Shabbos, since the opportunity to complete the offerings, which extended it to seven days, no longer exists, while Rosh Hashanah and Yom Kippur are like the festivals. The Chachamim reversed this: Shavuos is like the festivals, and Rosh Hashanah and Yom Kippur are like Shabbos. In practice the ruling is that all of these - Shavuos, Rosh Hashanah and Yom Kippur - interrupt the shivah.