Beitzah, Chapter Two, Mishnah 1. This Mishnah deals with cooking on Yom Tov for the needs of Shabbos, and with the dispensation known as eruv tavshilin.
The Mishnah opens: "Yom tov shechal lihyos erev Shabbos, lo yevashel adam betechilah beyom tov leShabbos" - when Yom Tov falls on Friday, a person may not cook on it at the outset for the needs of Shabbos.
The Gemara in Pesachim discusses the source of this prohibition. Simply understood, by Torah law it is permitted to cook on Yom Tov for Shabbos, and the Gemara there explains various reasons for this leniency; the Sages, however, forbade it unless one made an eruv tavshilin.
The Mishnah continues: "Aval ofeh umevashel hu leyom tov, ve'im hosir - hosir leShabbos" - he may bake and cook for the needs of Yom Tov, and if food is left over, he may use the leftovers for Shabbos.
Eruv Tavshilin:
Here the Mishnah presents the concept of eruv tavshilin: "Ve'oseh tavshil me'erev yom tov vesomech alav leShabbos" - a person prepares a cooked food before Yom Tov and relies on it in order to cook on Yom Tov for the needs of Shabbos. In practice, the custom is to make the eruv with a cooked food and with a baked item.
The Gemara brings two reasons for the enactment of eruv tavshilin:
For the honor of Shabbos: to ensure that a person sets aside food fitting for the honor of Shabbos. Yom Tov is an out of the ordinary day, and amid all its activity a person is liable to forget about Shabbos; once the Shabbos preparations begin already before Yom Tov, Shabbos is not forgotten, and the proper foods are set aside for it.
For the honor of Yom Tov: so that people will say, "If I may not cook on Yom Tov for the needs of Shabbos, all the more so may I not cook on Yom Tov for the needs of a weekday" - which is indeed forbidden, and the enactment reinforces this principle.
How many cooked foods are required for the eruv?
"Beis Shammai omrim shnei tavshilin" - in their view, two cooked foods are required.
"UBeis Hillel omrim tavshil echad" - in their view, one cooked food suffices.
"Veshavin bedag uveitzah she'alav shehen shnei tavshilin" - Beis Shammai and Beis Hillel agree that a fish and an egg cooked upon it count as two cooked foods, and therefore even according to Beis Shammai this suffices; and according to Beis Hillel, of course, only one cooked food is needed in any case.
If the Eruv Was Lost or Eaten:
If the eruv was eaten or lost before he began cooking for Shabbos, he may not cook on Yom Tov for Shabbos, since he has nothing on which to rely. The Gemara teaches, however, that if he began cooking and in the middle the eruv was lost or eaten, he may continue cooking.
"Im shiyer mimenu kol shehu, samach alav leShabbos" - even if most of it was lost and only a little remained, he relies on it for the needs of Shabbos. The measure of an eruv tavshilin, as explained in the Gemara, is a kezayis; and even when we say "if a bit of it remained," the intent is that at least a kezayis remains of it.
In summary: in this Mishnah we learned that one does not cook on Yom Tov at the outset for the needs of Shabbos, but one may bake and cook for the needs of Yom Tov, and whatever is left over may be used for Shabbos. We also learned the enactment of eruv tavshilin and its two reasons, the honor of Shabbos and the honor of Yom Tov; the dispute between Beis Shammai and Beis Hillel over the number of cooked foods and their agreement regarding a fish with an egg upon it; and the law of an eruv that was lost or eaten, and the required remaining measure - a kezayis.