Beitzah, Chapter Two, Mishnah Two. This mishnah deals with the law of immersion before Shabbos or on Shabbos itself, as preparation for Yom Tov.
The principle underlying the law:
A vessel that cannot be used because of its state of tumah - immersing it in a mikvah is comparable to repairing the vessel. Actually repairing a vessel physically is forbidden on Shabbos and Yom Tov, and the question is whether repairing it in the halachic sense is treated like an actual repair. Chazal ruled that rabbinically one may not do so.
The question in our mishnah revolves around whether this law applies to a person as well - a person who is tamei and changes his halachic status through immersion.
"Yom tov shechal lihyos achar haShabbos" - a Yom Tov that falls right after Shabbos:
When Yom Tov falls immediately after Shabbos, and there is a mitzvah to become pure and to purify oneself as preparation for Yom Tov, a difficulty arises: ordinarily a person would go to the mikvah on erev Yom Tov, but here erev Yom Tov is Shabbos itself.
"Beis Shammai omrim: matbilin es hakol milifnei haShabbos" - everything must be immersed before Shabbos, both vessels and people, since one may immerse neither a person nor vessels on Shabbos, and such immersion is like making a repair on Shabbos. Therefore, whoever wishes to be ready for Yom Tov must do so before Shabbos.
"UBeis Hillel omrim: keilim milifnei haShabbos, ve'adam baShabbos" - regarding vessels, Beis Hillel agree with Beis Shammai that they must be immersed before Shabbos, because rabbinically it looks like repairing them. A person, however, may immerse even on Shabbos, since it is not evident that he is coming to change his status, given that he could be immersing simply for his own enjoyment.
In summary: In this mishnah we learned that immersing a vessel that is tamei is considered repairing it, and it was therefore forbidden on Shabbos and Yom Tov. According to Beis Shammai, vessels and people are governed by one law, and everything requires immersion before Shabbos; according to Beis Hillel the laws differ: vessels - before Shabbos, while a person may immerse even on Shabbos, because it is not recognizable as a repair.