Before us is a Mishnah in Maseches Nedarim dealing with a person who takes a vow until a certain time. The starting point is that generally, when we say "until," the meaning is up to but not including - the stated time itself is not included in the vow, as we will see in our Mishnah.
The words of the Mishnah:
"Ad haPesach" - until Pesach: he is forbidden until it arrives. The one who vowed is forbidden only until Pesach comes, and once Pesach arrives the vow ceases to apply. We see that "until" means up to but not including: until Pesach, but not including Pesach itself.
"Ad she'yehei" - until it will be: he is forbidden until it departs. One who says "until it will be Pesach" means until the entire Pesach has been, and therefore he is forbidden even during Pesach itself until it passes and departs.
"Ad lifnei haPesach" - until before Pesach: here the Tannaim disagree. Rabbi Meir says he is forbidden until it arrives, that is, until the beginning of Pesach; and Rabbi Yosi says he is forbidden until it departs, until Pesach ends.
It is worth noting that some have the reading "ad p'nei haPesach" instead of "ad lifnei haPesach." And indeed, linguistically this is the source of the word "lifnei": ad p'nei, lifnei.
The basis of the dispute according to the Gemara in Kiddushin:
The Gemara in Maseches Kiddushin explains that the dispute hinges on the understanding of the word "p'nei":
According to Rabbi Meir: "P'nei haPesach" means the face of Pesach, that is, before Pesach - until Pesach begins.
According to Rabbi Yosi: "P'nei" comes from the word "yifneh," to turn away - until Pesach turns away, that is, until it ends.
It emerges that the difference depends on how one understands the word "p'nei": does it mean the face, the front, of the thing, or does it come from the sense of departing - until Pesach turns away and passes?
The basis of the dispute according to the Gemara in Nedarim:
However, the Gemara here in Nedarim says that in fact everyone agrees that "ad p'nei Pesach" means until the face of Pesach, and there is no intention at all of the sense of "yifneh." The difficulty is that as long as any part of Pesach remains - even at its final moment - the person is still standing before some part of Pesach.
Therefore the dispute revolves around a fundamental question: does a person who takes a vow include in his vow something that is in doubt, or does he not wish to include any doubt in it?
Rabbi Meir: A person does not want to include in his vow something that will be subject to doubt, and therefore when he says "ad p'nei haPesach" he certainly means the first possible interpretation - the very beginning of Pesach. Even though more of Pesach remains and its continuation too could be called "p'nei haPesach," the intent of the one who vowed is only until the first possible moment.
Rabbi Yosi: A person does bring himself into doubt and holds on to the doubt in his words at the time of his vow, and therefore he is forbidden until the last moment of Pesach, since any part of Pesach can be understood as "ad p'nei Pesach."
In summary: In this Mishnah we learned that "ad haPesach" means up to but not including, "ad she'yehei" means until all of Pesach has departed, while regarding the wording "ad lifnei haPesach" Rabbi Meir and Rabbi Yosi disagree. The Gemara in Kiddushin made their dispute depend on the understanding of the word "p'nei" - the face of Pesach, or from the word "yifneh"; whereas the Gemara here in Nedarim explained that everyone agrees on the meaning of the word, and the dispute is whether a person brings himself into doubt with his vow - according to Rabbi Meir he does not, and according to Rabbi Yosi he does.