Eduyos, Chapter 4, Mishnah 8. This mishnah and the one that follows deal with matters of yibum, so we will begin with a short and only partial introduction to the subject, so that everything will be clear.
A Short Introduction to the Laws of Yibum:
When a man dies without leaving living descendants, his wife, who is now a widow, is bound by a mitzvah that can be fulfilled in one of two ways:
Yibum - marrying her brother-in-law, her husband's surviving brother, if there is one.
Chalitzah - a procedure that includes removing the brother-in-law's shoe, spitting on the ground, and declaring that he refuses to build up his brother's name.
Until one of these two is done, the widow may not marry anyone else. She remains bound to the surviving brother-in-law, and this bond is called 'zikah' - she is tied to him and obligated to him in a matter that has not been completed, until the zikah is released through yibum or through chalitzah, which is a kind of divorce.
Under ordinary circumstances, marrying a brother's wife is forbidden by the Torah under the prohibition of 'eishes ach', which carries the penalty of kareis - a severe punishment and an absolute prohibition. This prohibition is set aside by the mitzvah of yibum. However, the other restrictions of forbidden relationships are not set aside by the mitzvah of yibum. Therefore, if there is an additional reason why the yevamah is forbidden to him, beyond the fact that she is his brother's wife, that man remains under his prohibition and there is no mitzvah of yibum here at all.
Our Case:
Imagine two brothers, Reuven and Shimon. Reuven is married to Shoshanah, who is Shimon's daughter - in other words, Reuven married his niece. If Reuven dies, Shimon certainly cannot perform yibum with Shoshanah, since she is his own daughter, and on this everyone agrees.
However, according to Torah law a man could be married to two wives. Reuven may have been married to Shoshanah, his niece, and also to a second wife, Rachel, who is not related to anyone. And here is the question of the mishnah: if Shimon cannot marry his own daughter, who is one of Reuven's widows, may he marry the second widow, Rachel? Or is this a 'package deal' - since Shoshanah is exempt from any zikah or bond with Shimon, her co-wives are exempt along with her, and no bond remains between Rachel and Shimon.
The relationship between the wives of one man is called in the language of the mishnah 'tzaros', each one being the tzarah of the other. The question is whether the exemption of one co-wife from yibum and chalitzah exempts all of the co-wives. This is the question our mishnah comes to answer.
The Dispute Between Beis Shammai and Beis Hillel:
"Beis Shammai matirin es hatzaros la'achim" - according to Beis Shammai, the surviving brother, Shimon, even though he cannot marry Shoshanah because she is his daughter, may marry her co-wife, Rachel. And since it is permitted, it becomes an obligation: Shimon is required to release the zikah, either through yibum or through chalitzah, in order to allow Rachel to marry someone else. Until that is done, Rachel is restricted and is not free to marry anyone she wishes, but is bound to yibum or chalitzah with her brother-in-law Shimon.
"U'Beis Hillel osrin" - according to Beis Hillel this is not so; rather, they are all one bundle: if one co-wife among the widows is exempt from yibum and from chalitzah with a particular brother, all the co-wives are exempt. In our case, just as Shimon cannot perform yibum or chalitzah with his daughter Shoshanah, so too he cannot perform yibum or chalitzah with her co-wife, the second widow, Rachel.
The basis for this is a derivation from a verse in Vayikra (18:18), which forbids a man to marry a woman and her sister as co-wives to one another. That word appears superfluous, since the Torah could simply have written that it is forbidden to marry a woman and her sister as two wives; from here Beis Hillel derived that the co-wives are exempt in this manner as well.
And what if chalitzah or yibum was actually performed?
Here the results diverge sharply between the two houses. If Shimon performed chalitzah with Rachel - she removes his shoe, spits on the ground and declares, "So shall be done to the man who will not build up his brother's house":
Beis Shammai: this was exactly the right thing to do, and from now on Rachel is permitted to marry anyone - but not a kohen. A kohen is restricted from marrying certain categories of women, among them a zonah and a divorcee. Although Rachel is not actually divorced but is a chalutzah - a woman who was rejected from the bond because Shimon did not wish to marry her - a chalutzah is considered rabbinically equivalent to a divorcee. A divorcee is forbidden to a kohen by Torah law, and a chalutzah is forbidden to a kohen by rabbinic law, and therefore Beis Shammai disqualify her from the kehunah.
Beis Hillel: the chalitzah that Shimon performed with Rachel was an empty act with no substance whatsoever - like someone performing chalitzah with a random woman on the street. Since there was no need at all for her chalitzah and the act carries no consequence, she is not considered a chalutzah, and she is not like a divorcee by rabbinic law. It turns out that she wasted her time and effort, and she is permitted to marry anyone just as before, even a kohen.
And on the other side of the coin, yibum - Rachel moves in with Shimon and they live together as husband and wife:
"Nisyabmu - Beis Shammai machshir" - according to Beis Shammai this is the mitzvah of yibum performed properly, and everything has fallen into place.
"U'Beis Hillel poslin" - according to Beis Hillel there is nothing here but the Torah prohibition of a brother's wife. Since there was no bond and no obligation, Shimon has no permission whatsoever to marry his late brother's wife, and this is a prohibition carrying kareis and a completely forbidden marriage. The children born to this family are mamzerim; and the woman, Rachel, who lives with Shimon in a relationship forbidden by Torah law, becomes a "zonah" - a halachic term applying to any Jewish woman who has relations with a man forbidden to her - and from now on she is forbidden to marry a kohen.
It emerges, then, that on this matter Beis Hillel is the stringent one: they exempt the co-wives and forbid all of these acts from the outset.
The coexistence of Beis Shammai and Beis Hillel:
If we think about it, the family trees of the community of Beis Hillel and the community of Beis Shammai become mutually exclusive. For in the world of Beis Hillel, women who are considered fit are in fact zonos and their children mamzerim; and similarly in the world of Beis Shammai, women who required chalitzah or yibum marry other men and go out into the marketplace. The family trees are therefore hopelessly tangled, and one might have thought that the two houses would say to one another: the members of your community are forbidden to us, since by our standards you do not follow the rules.
But the Mishnah teaches that this was not the case: "Af al pi she'eilu poslin v'eilu machshirin, lo nimne'u Beis Shammai mi'lisa nashim mi'Beis Hillel, v'lo Beis Hillel mi'lisa nashim mi'Beis Shammai". This does not mean that they looked the other way or ignored the problems. They trusted one another: Beis Shammai would tell Beis Hillel that there was an outstanding young woman in their community, but that according to Beis Hillel's own position she should not be married because she is restricted as a mamzeres or the like, even though in their own view she was fit - and the same in reverse. They were trustworthy with one another to tell the truth and to present the halachic status of the women and men according to the other side's point of view, and out of this trustworthiness they were able to marry into one another's families. This expresses the importance of understanding that a disagreement in opinion need not lead to a rupture in the love and respect within the community.
And more than that: "V'chol hataharos v'hatuma'os she'hayu eilu m'taharin v'eilu m'tam'in - lo nimne'u mila'asos taharos eilu al gav eilu" - the same line of thinking applies to matters of tumah and taharah as well. Beis Hillel and Beis Shammai disagreed in countless places in these areas, and each community held a different view as to whether a given item was tamei or tahor, and nevertheless they did not refrain from eating in one another's homes, nor did they limit themselves from lending one another vessels for cooking, for preparing food, and the like, because they trusted one another. Beis Hillel trusted Beis Shammai to tell them what was permitted and what was forbidden according to their approach, and vice versa - and out of this they cultivated mutual respect, cooperation and harmony within the community.
In summary: in this Mishnah we learned the dispute between Beis Shammai and Beis Hillel regarding the law of the co-wives in yibum: Beis Shammai permit the co-wives to the brothers, while Beis Hillel exempt them all based on the derivation from the verse in Vayikra. From this flow the consequences with regard to chalitzah (a chalutzah who is forbidden to a kohen by rabbinic law, as opposed to a meaningless, empty act) and with regard to yibum (a properly fulfilled mitzvah of yibum, as opposed to the prohibition of a brother's wife, mamzerus and the status of a zonah). And the Mishnah closed with an eternal lesson: even though the two houses held different worldviews and related to the practical halachah in entirely different ways, they continued to live together, to marry one another and to be good neighbors without any problem at all - and it is for this that this Mishnah became so famous.
It is worth noting that this Mishnah does not appear here alone, but also in Maseches Yevamos, chapter 1, Mishnah 4. In the next Mishnah we will continue with the topics of yibum.