Shabbos, Chapter 15, Mishnah 3. This Mishnah deals first with the limitations on folding garments on Shabbos, and then with the relationship between the sanctity of Shabbos and the sanctity of Yom Kippur. The Mishnah opens: "Mekapelin es hakeilim" - one may fold garments along their creases, "afilu arba'ah vachamishah pe'amim" - even four or five times.
The Conditions for Folding on Shabbos:
The Gemara qualifies this permission with certain conditions. First, the person must intend to wear the garment again on that same Shabbos, for otherwise he is preparing on Shabbos for the weekdays. Second, folding involves a concern of "fixing" the garment, and therefore four conditions are required:
One person only: The folding must be done by a single person and not by two together, since folding done by two people constitutes an added measure of "fixing."
New garments: In a new garment the crease is not significant, whereas in older garments folding along the creases is considered "fixing."
White garments: Colored garments, however, may not be folded along their creases.
He has no other garment to change into: He specifically needs this particular garment.
For this reason the common practice on Shabbos is not to fold the tallis along its creases, but rather off its creases, and only after Shabbos is it folded in the usual manner.
"Umatzi'in es hamitos":
The Mishnah continues: "Umatzi'in es hamitos" - one may make the beds, spreading out the sheets and arranging them. This does not refer to routine bed-making, but to actually changing the sheets, which is a more significant act. This permission applies "mileil Shabbos leShabbos" - from Friday night for the needs of Shabbos itself, "aval miShabbos u'motza'ei Shabbos" - but if the sheets need to be changed for after Shabbos, this may not be done on Shabbos.
The Dispute Between Rabbi Yishmael and Rabbi Akiva:
Rabbi Yishmael says: "Mekapelin es hakeilim, matzi'in es hamitos, miYom HaKippurim leShabbos" - both of these acts, folding garments and making the beds, may be done on Yom Kippur for the needs of Shabbos, because the sanctity of Shabbos is greater than the sanctity of Yom Kippur, and therefore one may prepare on Yom Kippur for the needs of Shabbos.
Rabbi Yishmael brings a further proof from the laws of the offerings in the Temple: the fats of Shabbos are offered on Yom Kippur. The limbs left over from the Shabbos offerings may be burned on Yom Kippur when it falls out on Motza'ei Shabbos - a situation that does not occur in our fixed calendar, but which could happen in the days when the month was established by testimony of witnesses. Here too the reason is that Shabbos is weightier than Yom Kippur. The Mishnah used the term "fats," but in practice this refers to all the limbs of the daily tamid offering, for since it is an olah, all of it is burned; and whatever was not burned on Shabbos may be burned on Yom Kippur.
Rabbi Akiva disagrees and holds that the offerings of Shabbos are not brought on Yom Kippur, and likewise those of Yom Kippur are not brought on Shabbos. As for the prohibition of offering those of Yom Kippur on Shabbos - even Rabbi Yishmael agrees to that. Consequently, Rabbi Akiva holds that one may not fold garments or make the beds on Yom Kippur for the needs of Shabbos, and certainly not the reverse.
In summary: In this Mishnah we learned that one may fold garments on Shabbos, even several times, provided the conditions are met: an intention to wear them on that same Shabbos, folding by one person, new and white garments, and having no other garment to change into. We also learned that the beds may be made on Friday night for the needs of Shabbos but not for the needs of Motza'ei Shabbos, and about the dispute between Rabbi Yishmael and Rabbi Akiva as to whether the greater sanctity of Shabbos permits preparing on Yom Kippur for the needs of Shabbos, both regarding folding and bed-making and regarding the burning of the limbs in the Temple.