Commentary for Rosh Hashanah 1:3
עַל שִׁשָּׁה חֳדָשִׁים הַשְּׁלוּחִין יוֹצְאִין, עַל נִיסָן מִפְּנֵי הַפֶּסַח, עַל אָב מִפְּנֵי הַתַּעֲנִית, עַל אֱלוּל מִפְּנֵי רֹאשׁ הַשָּׁנָה, עַל תִּשְׁרֵי מִפְּנֵי תַקָּנַת הַמּוֹעֲדוֹת, עַל כִּסְלֵו מִפְּנֵי חֲנֻכָּה, וְעַל אֲדָר מִפְּנֵי הַפּוּרִים. וּכְשֶׁהָיָה בֵית הַמִּקְדָּשׁ קַיָּם, יוֹצְאִין אַף עַל אִיָּר מִפְּנֵי פֶסַח קָטָן:
For six months messengers go out. [When beth-din sanctified the New Moon, they sent out (messengers) to inform the exile which day they had sanctified, whether the thirtieth, so that the month past was defective; or the thirty-first, so that the month past was full. (And the messengers of beth-din may desecrate neither Shabbath nor yom tov.)]: For Nissan, as it affects (the date of) Pesach; for Av, as it affects the fast (of the ninth of Av) [more afflictions having befallen us then than on (the dates of) the other fasts.]; for Elul, as it affects Rosh Hashanah [They apprise us when it is Elul, and Rosh Hashanah is observed on the thirtieth of Elul in the exile. For on most years Elul is not intercalated. And even though there is the possibility that beth-din might intercalate it, they (those in the exile) have no way of knowing this, so that, perforce, they must follow the majority of the years. And if they did not know when Elul began, they would not know its thirtieth day.]; for Tishrei, as it affects proper institution of the festivals. [The messengers go out the day after beth-din sanctified Tishrei, and go as far as they can until the festival, apprising them whether or not beth-din had intercalated Elul, so that they not be apprehensive on Yom Kippur and Succoth (as to whether they are observing them in their proper times)]; for Kislev, as it affects Channukah; and for Adar, as it affects Purim. And when the Temple stood, they would also go out for Iyyar, as it affects "the minor Pesach" [Pesach Sheni].
Bartenura on Mishnah Rosh Hashanah
English Explanation of Mishnah Rosh Hashanah
On Nisan because of Pesah;
On Av because of the fast.
On Elul because of Rosh Hashanah.
On Tishri because of the setting of the festivals.
On Kislev because of Hanukah.
And on Adar because of Purim.
When the Temple stood, they used also to go out to report Iyar because of Pesah Katan (Pesah.
In order to know what date holidays would fall on, people would need to know on what day the previous month had begun. This would be simple for those living in and near Jerusalem because they could immediately find out whether the court had decreed the thirtieth day of the previous month to be Rosh Hodesh. Our mishnah teaches that they used to send out messengers from Jerusalem so that people who lived further away could also find out when Rosh Hodesh had been decreed. However, they did not send these messengers out every month of the year, only on months which contained holidays.
Most of this mishnah is self-explanatory. Note that messengers do not need to go out for Shavuot because once one knows when Pesah falls, the date of Shavuot is known automatically.
Section two: “The fast” refers to Tisha B’av, the Ninth of Av.
Section three: Messengers would go out on Elul so that people would be able to celebrate Rosh Hashanah on the thirtieth day of Elul, lest that day turns out to be the first of Tishri. They would celebrate Rosh Hashanah on the next day as well, lest Elul has thirty days.
Section four: On Tishri the messengers would go out so that people would know when Yom Kippur and Sukkot fall.
Section seven: Pesah Sheni was the second chance for people to offer the pesah sacrifice if they did not offer it the first time (see Numbers 9:10-11 and Mishnah Pesahim 9:1). After the destruction of the Temple this holiday lost any practical relevance. We should note that by noting that when the Temple stood messengers went out for Pesah Sheni the mishnah also teaches that for all of the rest of the holidays the messengers still go out. After the destruction of the Temple only Pesah Sheni has lost its relevance.