The Mishnah before us discusses matters forbidden by rabbinic decree on Shabbos and Yom Tov, and divides them into three categories:
Because of shevus - something the Sages forbade, which contains no element of mitzvah in itself.
Because of reshus - something that contains a measure of mitzvah, though it is not an absolute obligation, and nevertheless the Sages forbade it.
Because of mitzvah - matters that are full-fledged mitzvos, and even so were forbidden by rabbinic decree.
The words of the Mishnah:
"Kol shechayavin alav mishum shevus, mishum reshus, mishum mitzvah - b'Shabbos, chayavin alav b'Yom Tov" - anything for which one is liable because of shevus, because of reshus, or because of mitzvah on Shabbos, one is liable for it on Yom Tov as well. The term "liability" usually refers to liability for a korban or lashes, but here it means only a rabbinic prohibition. In other words: anything forbidden under one of these three categories on Shabbos may not be done on Yom Tov either.
And these are because of shevus:
"Lo olin ba'ilan" - one may not climb a tree, lest he detach leaves or branches and transgress the melachah of detaching.
"V'lo rochvin al gabei behemah" - one may not ride an animal, lest he break off a branch to strike the animal with it.
"V'lo shatin al p'nei hamayim" - one may not swim on the water, lest he come to fashion a sort of vessel, a float.
"V'lo metapchin, v'lo mesapkin, v'lo merakdin" - one may not clap the hands, one may not slap hand on thigh, and one may not dance (and for this reason it is customary to clap with a change from the usual manner). All three were forbidden out of concern that one might come to repair a musical instrument.
In all of these there is no element of mitzvah whatsoever, and therefore they fall under the category of "shevus" alone.
And these are because of reshus:
Matters that carry no absolute obligation, but do contain a measure of mitzvah:
"Lo danin" - one may not sit in judgment. This is not a full-fledged mitzvah, since the case is one where there are men in the city greater than he who are available to judge, and the matter does not rest entirely upon him.
"V'lo mekadshin" - one may not betroth a woman, even though there is a mitzvah in doing so. The case is one who is already married and has a wife and children, and the obligation of the mitzvah itself does not rest upon him.
"V'lo choltzin" - one may not perform chalitzah, since the case is one where there is an older brother who is able to do it.
"V'lo meyabmin" - one may not perform yibum with his brother's widow, for the same reason: there is another brother, older than he, who can fulfill it.
And these are because of mitzvah:
Matters that are forbidden even though they are full-fledged mitzvos:
"Lo makdishin" - one may not consecrate an item to hekdesh on Shabbos or Yom Tov.
"V'lo ma'arichin" - one may not make a vow of arachin, in which the Torah fixed a set value for people according to age and gender.
"V'lo macharimin" - one may not declare a cherem, which is another type of vow.
"V'lo magbihin Terumah uma'aser" - one may not separate Terumah and ma'aser on Shabbos or Yom Tov.
The reason for the prohibition: hekdesh, arachin and charamim resemble commercial dealings, and buying and selling were forbidden lest one come to write. Separating Terumah and ma'aser was forbidden because it appears like repairing a vessel.
"Kol eilu b'Yom Tov amru, kal vachomer b'Shabbos" - all of these were stated as prohibitions regarding Yom Tov, and all the more so are they forbidden on Shabbos.
The Mishnah concludes: "Ein bein Yom Tov l'Shabbos ela ochel nefesh bilvad" - there is no difference between Yom Tov and Shabbos except for the work of preparing food, such as cooking and the labors connected with preparing food, which are permitted on Yom Tov and forbidden on Shabbos.
The view of Beis Shammai and the view of Beis Hillel:
The Gemara explains that this conclusion of the Mishnah follows the view of Beis Shammai, who hold that only labors performed literally for the sake of food preparation were permitted. Beis Hillel, however, hold, as we learned earlier in the tractate, that since the labors of carrying, kindling and cooking were permitted for the sake of food preparation, they were permitted for other needs as well - as long as they involve a need of a mitzvah or of enjoyment, even though they are not for the sake of food preparation.
For example: heating water for washing one's feet has nothing to do with food preparation. According to Beis Shammai this is forbidden, and according to Beis Hillel it is permitted. It emerges that the wording of the Mishnah, "ochel nefesh alone," meaning solely for the purpose of eating, follows the view of Beis Shammai and not the view of Beis Hillel.
In summary: in this Mishnah we learned three categories of rabbinic prohibitions that apply equally on Shabbos and Yom Tov: because of shevus (climbing a tree, riding an animal, swimming, clapping, slapping and dancing), because of reshus (judging, betrothal, chalitzah and yibum - in a case where there is no absolute obligation), and because of mitzvah (hekdesh, arachin, charamim and separating Terumah and ma'aser). We also examined the conclusion of the Mishnah, that there is no difference between Yom Tov and Shabbos except for food preparation alone, and that this conclusion follows the view of Beis Shammai, while Beis Hillel permit additional needs of mitzvah and enjoyment as well.