Sukkah, Chapter 4, Mishnah 1. This chapter deals with the various mitzvos of the festival of Sukkos and the days on which each of them applies, and this opening mishnah serves as an introduction to the entire chapter and to all the mitzvos discussed in it.
The mishnah lists the mitzvos and the number of days for each:
"Lulav va'aravah - shishah v'shivah" - the mitzvah of taking the four species (lulav, esrog, hadassim and aravos), and also the mitzvah of the aravah that was practiced in the Beis HaMikdash on all the days of the festival. In some years these mitzvos are performed for six days and in other years for seven, and as we will see, this depends on which day Shabbos falls out: whether it falls on the first day of the festival or not.
"Hallel u'simchah - shemonah" - the mitzvah of reciting Hallel and the mitzvah of rejoicing, whose essence is eating the meat of the shelamim offering, as the saying goes, "there is no rejoicing except with meat" - the meat of shelamim. This was the primary form of the mitzvah of rejoicing in the days of the Beis HaMikdash. Both of these mitzvos apply on all eight days, including Shemini Atzeres.
"Sukkah v'nisuch hamayim - shivah" - the mitzvah of dwelling in the sukkah, and the pouring of the water on the altar, which was a special mitzvah for the days of Sukkos in the Beis HaMikdash. Both of these mitzvos apply only on the seven days of the festival, and not on Shemini Atzeres.
"V'hechalil - chamishah v'shishah" - the playing of the flute in fact refers to many musical instruments, but the flute was the primary instrument used at the Simchas Beis HaSho'evah. This celebration took place for either five days or six days, and here too it depends on when Shabbos falls out, since the Simchas Beis HaSho'evah was held only on Chol HaMo'ed and not on Shabbos or Yom Tov.
In summary: This mishnah opens the chapter with a general introduction, listing the mitzvos of the festival according to the number of days on which each applies: lulav and aravah - six or seven; Hallel and rejoicing - eight; sukkah and the water libation - seven; and the flute - five or six. In the remainder of the chapter these mitzvos will be explained one by one.