Moed Katan, Chapter Three, Mishnah Six
The festivals which most clearly put an end to shivah and shloshim are Pesah and Sukkot since they are both seven or if you include Shmini Atzeret (the last day of Sukkot), eight days long. In contrast, Shavuot, Rosh Hashanah and Yom Kippur last only one day. Further complicating the matter, Rosh Hashanah and Yom Kippur are holidays, but not actually festivals. Festival in Hebrew (regel) refers only to the three pilgrimage holidaysPesah, Shavuot and Sukkot, when one was supposed to visit the Temple. Due to these complications, our mishnah contains a debate over where these one day holidays are treated like Shabbat or like Pesah and Sukkot.
1) Rabbi Eliezer says: From the time the Temple was destroyed, Atzeret (Shavuot) is like Shabbat.
2) Rabban Gamaliel says: Rosh Hashanah and Yom Kippur are like festivals.
3) The sages say: [the rule is] not according to the words of this one nor that one, rather Atzeret is like the festivals and Rosh Hashanah and Yom Kippur are like Shabbat.
Section one: Atzeret is the word used in the Mishnah to refer to Shavuot. When the Temple still stood, Atzeret was similar to the other festivals. One who did not bring the appropriate sacrifice on Atzeret itself could bring it for the following six days. Hence, in a sense Atzeret was a seven day holiday, even though it was only fully observed for one day. When the Temple still stood, it would interrupt mourning. Once the Temple was destroyed and sacrifices could no longer be brought, Atzeret ceased being a seven day holiday and hence is treated like Shabbat when it comes to mourning.
Section two: According to Rabban Gamaliel Rosh Hashanah and Yom Kippur are like festivals. Assumedly, his reasoning is that anything that is not Shabbat counts as a festival. He would therefore disagree with Rabbi Eliezer. Today the halakhah follows Rabban Gamaliel. Only Shabbat does not interrupt mourning.
Section three: The other sages disagree with both Rabban Gamaliel and Rabbi Eliezer. The category of festivals includes all pilgrimage holidays, and even Atzeret after the destruction of the Temple. Rosh Hashanah and Yom Kippur are not festivals and hence they function like Shabbat. The sages opinion seems to be a literal interpretation of the last clause of yesterdays mishnah which stated that festivals interrupt. The sages interpret this to mean only festivals and not other holidays.