Berachos Chapter 8, Mishnah 1. The Mishnah opens with an introductory statement that defines the character of the entire chapter: "Elu devarim shebein Beis Shammai u'Beis Hillel bis'udah" - the items brought in the coming mishnayos are the details in which Beis Shammai and Beis Hillel disagreed regarding the laws of the meal, that is, the laws governing how a meal is eaten.
The first topic - the order of the berachos of Kiddush:
On Friday night and on the night of Yom Tov we make Kiddush, and it contains two berachos:
The berachah of "Borei pri hagafen" over the wine.
The berachah of Kiddush - "mekadesh haShabbos" on Shabbos, or "mekadesh Yisrael v'hazemanim" on Yom Tov.
The question is: which of the berachos comes first - the berachah over the wine or the berachah of Kiddush? In practice we recite the berachah over the wine first, since we rule like Beis Hillel, but let us look closely at the wording of the Mishnah itself.
"Beis Shammai omrim: mevarech al hayom v'achar kach mevarech al hayayin" - one first recites the berachah of the day ("mekadesh haShabbos" or "mekadesh Yisrael v'hazemanim"), and afterwards the berachah over the wine. Their reasoning, as the Gemara explains: the day is what brings the wine to the table - the entire institution of reciting Kiddush over wine stems from the sanctity of the day. Therefore one recites the berachah of the day first, and only afterwards, before drinking the wine, recites "Borei pri hagafen".
"Beis Hillel omrim: mevarech al hayayin v'achar kach mevarech al hayom" - first the berachah of "Borei pri hagafen" over the wine, and afterwards the berachah of "mekadesh haShabbos" or "mekadesh Yisrael v'hazemanim". There are two reasons behind their position: the wine is what enables the Kiddush, for without it one could not make Kiddush at all; and furthermore, because of the rule that when one item is frequent and another is not, the more frequent comes first: the berachah over wine is recited more often than the berachah of "mekadesh haShabbos", and therefore it comes first.
In summary: This Mishnah opens the chapter by presenting the matters in which Beis Shammai and Beis Hillel disagreed regarding the laws of the meal, and it brings first their dispute about the order of the berachos of Kiddush: according to Beis Shammai the berachah of the day comes first, since the day is what brings the wine; and according to Beis Hillel, which is the accepted halachah, the berachah over the wine comes first - because the wine is what enables the Kiddush, and because it is the more frequent berachah.