Megillah, Chapter 1, Mishnah 5. This mishnah, along with the seven mishnayos that follow until the end of the chapter, does not deal with Purim or with the Megillah. They were placed here because the previous mishnah concluded with the phrase "ein bein": there is no difference between the first Adar and the second Adar except for the reading of the Megillah and gifts to the poor. Along these same lines, these mishnayos discuss pairs of subjects that are very similar to one another, sharing a broad common denominator except for one detail that they do not share. They all open with the words "ein bein" - there is no difference between this and that except - and for that reason they are known as the "ein beinayim."
"Ein bein Yom Tov l'Shabbos":
The language of the mishnah: "Ein bein Yom Tov l'Shabbos ela ochel nefesh bilvad" - the difference between Yom Tov and Shabbos is that on Yom Tov melachah is permitted for the sake of preparing food, which is not the case on Shabbos. There are indeed other differences between them, but this is one primary difference.
The Gemara explains that this wording follows specifically the opinion of Beis Shammai. In Maseches Beitzah, Beis Shammai and Beis Hillel disagree about whether the melachos that were permitted for the sake of preparing food were also permitted for purposes other than preparing food:
Beis Hillel: "Mitoch" - since these melachos were permitted for the sake of preparing food, those same melachos were permitted even for a purpose that is not preparing food, as long as they are done for the sake of one's enjoyment of Yom Tov.
Beis Shammai: They were permitted only for the sake of preparing food.
Since the mishnah says "ela ochel nefesh bilvad" - for the sake of preparing food and for nothing else - it emerges that its words follow the opinion of Beis Shammai and not the opinion of Beis Hillel.
"Ein bein Shabbos l'Yom HaKippurim":
The mishnah continues with another "ein bein": "Ein bein Shabbos l'Yom HaKippurim ela shezeh zedono bidei adam v'zeh zedono b'kareis":
Shabbos: One who desecrates it intentionally - his punishment is at the hands of man, meaning death at the hands of beis din.
Yom Kippur: One who desecrates it intentionally - he is liable to kareis, which is a punishment carried out by Heaven and not by man.
In summary: In this mishnah we began the series of "ein beinayim" that concludes the chapter, taught here as a continuation of the wording of the previous mishnah. We learned two differences: between Yom Tov and Shabbos - the permission to perform melachah for the sake of preparing food, and the wording of the mishnah follows the opinion of Beis Shammai; and between Shabbos and Yom Kippur - that the intentional violation of the one is punished at the hands of man, and the intentional violation of the other by kareis.