***Please note that this Halacha was presented last year, in 5774 (2014), when Rosh Hashanah fell on Thursday and Friday***
The Tur (Rabbenu Yaakob Ben Asher, 1269-1343) observes (in Orah Haim, Siman 428) that the Sages established the calendar in such a way that the first day of Rosh Hashanah cannot fall out on Sunday, Wednesday or Friday. This principle is known by the phrase, "Lo Adu Rosh" – Rosh Hashanah cannot begin on the days represented by the letters Alef (Sunday), Dalet (Wednesday) and Vav (Friday).
The reason why the Sages did not want Rosh Hashanah to begin on Sunday is because this would result in Hoshana Rabba falling on Shabbat, which would prevent us from fulfilling the special Misva of Araba on that day. In order to ensure that we would be able to observe this precious Misva, the Sages saw to it that Hoshana Rabba would never fall on Shabbat. And if Rosh Hashanah would fall on Wednesday, then Yom Kippur would fall on Friday. This would result in two consecutive days – Friday and Shabbat – when burial is forbidden, and if somebody would die, Heaven forbid, the body would decompose over the course of the two days. In order to protect the dignity of those who pass away at the beginning of Yom Kippur, the Sages ensured that Yom Kippur would never fall on Friday. For the same reason, they wanted to ensure that Yom Kippur would never fall out on Sunday – which would again result in two consecutive days when burial is forbidden – and thus Rosh Hashanah cannot begin on Friday, which would result in Yom Kippur falling on Sunday.
The day when Rosh Hashanah begins affects the Torah reading of Parashiyot Nisavim and Vayelech. When Rosh Hashanah begins on Monday or Tuesday, then Parashat Nisavim is read alone on the Shabbat before Rosh Hashanah. (It should be noted that the preceding Parasha, Parashat Ki-Tabo, is always read before Rosh Hashanah, as it contains the "Kelalot" ("curses"), which we want to keep in the previous year and not bring with us into the new year.) Parashat Vayalech is then read on Shabbat Shuba – the Shabbat in between Rosh Hashanah and Yom Kippur – and Parashat Haazinu is read on the Shabbat in between Yom Kippur on Sukkot. The final Parasha, Parashat Vezot Ha’beracha, is, of course, always read on Simhat Torah. But when Rosh Hashanah begins on Thursday or Shabbat, then Parashiyot Nisavim and Vayelech are read together on the Shabbat before Rosh Hashanah, and Parashat Haazinu is read on Shabbat Shuba. In such situations, there is no Shabbat in between Yom Kippur and Sukkot, and we must therefore read Parashat Nisavim and Parashat Vayelech together on the same Shabbat. The Shulhan Aruch writes that this rule is known by the phrase, "Bag Ha’melech Pat Va’yelech." This means that if "Ha’melech" – the holiday of Rosh Hashanah, in which we focus on the theme of divine kingship – begins on Monday or Tuesday ("Bag"), then we "separate" ("Pat," from the word "Petitim," or "pieces") Parashat Vayelech.
