Welcome to Seder Nashim and to Maseches Yevamos. We begin with chapter 1, mishnah 1. To understand the mishnah, we first need the basic principle of the mitzvah of yibum: when a man dies without leaving children, his wife falls to yibum, and she must undergo yibum with her late husband's brother.
An ervah who exempts from yibum:
A brother's wife is forbidden by Torah law, and only in a case where the brother left no children was she permitted for the sake of yibum. However, if that widow is also an ervah with respect to the surviving brother, meaning a woman who is forbidden to him in her own right, he cannot perform yibum with her.
For example: the brother who died had married his brother's daughter. When he died, his widow should fall to yibum before his brother, except that this woman is the daughter of the surviving brother. Since another prohibition of ervah rests upon her, he cannot perform yibum with her, and she is exempt from the mitzvah of yibum. She does not even require chalitzah, which is the alternative to yibum.
A tzarah and the tzarah of a tzarah:
Suppose the deceased brother was married not only to the daughter of the surviving brother but also to another woman, who is called a 'tzarah' (co-wife). If one of the widows of the man who died childless was an ervah, the tzarah is exempt as well, and this is derived from a verse.
And not only that: if that tzarah underwent yibum with a third brother, who has no ervah connections in this case and is not married to the brother's daughter, and that brother has another wife, and he dies, then the other wife is called the 'tzarah of a tzarah', and she too is exempt. And so it continues on and on without end, all according to the number of brothers.
Our mishnah lists fifteen women who can be an ervah, and because of this they themselves are exempt from yibum, and so are their tzaros, the women married along with them to the same husband, and the tzaros of their tzaros, and so on.
The words of the mishnah:
"Chamesh esrei nashim potros tzaroseihen v'tzaros tzaroseihen min hachalitzah u'min hayibum ad sof haolam" - these fifteen women exempt, meaning that they cause no obligation of yibum or chalitzah to take effect upon their tzaros and upon the tzaros of their tzaros, without end, all according to the number of brothers.
And these are they:
Bito - the daughter of the surviving brother.
Bas bito - his daughter's daughter.
Bas beno - his son's daughter.
These three refer to descendants born to him not from his wife, but from a woman with whom he had a chance relationship and who bore a child from it. Even so, they are an ervah to him.
Bas ishto - his wife's daughter.
Bas benah - her son's daughter.
Bas bitah - her daughter's daughter.
These three, although they are not his biological children but rather his wife's descendants, are also an ervah to him.
Chamoso - his mother-in-law, his wife's mother.
Em chamoso - his mother-in-law's mother.
Em chamiv - his father-in-law's mother.
Achoso me'imo - his sister from his mother's side, and not from his father's side. The obligation of yibum applies only toward a brother from the father, so the case is that a paternal brother married the surviving brother's maternal sister, a marriage that is permitted to him, since she is not his relative but rather the sister of his brother from her mother. But once he dies, the surviving brother cannot perform yibum with her, since she is his sister.
Achos imo - his mother's sister.
Achos ishto - his wife's sister.
Eshes achiv me'imo - the wife of a brother who is his brother only from his mother's side. After she was widowed from him she married his paternal brother, and when that brother died he cannot perform yibum with her, since she is his yevamah by virtue of another brother, from his mother's side.
Eshes achiv shelo hayah b'olamo - the wife of a brother who did not live in the world at the same time as he did, such as where the surviving brother was born after his brother's death. The Torah requires that the two brothers live together in the world, and therefore one who was born afterward is not fit for yibum, and his brother's wife is a complete ervah to him.
Kalaso - his daughter-in-law, his son's wife.
All of these, in the words of the mishnah, exempt their tzaros and the tzaros of their tzaros from chalitzah and from yibum until the end of the world, according to the number of brothers.
Cases in which the tzarah is permitted:
In all these arrangements of tzaros, where the brother died leaving one wife who is an ervah toward his brother and another wife who is her tzarah, there are cases in which the tzaros are permitted:
Mesah: the ervah died before her husband died.
Mi'anah: her betrothal to the deceased brother was rabbinic, and she performed mi'un, which nullifies a rabbinic betrothal.
Nisgarshah: she was divorced from his brother.
Nimtzeis ailonis: she was found to be a woman incapable of bearing children.
In all these cases the tzaros are permitted, since they are not the tzarah of an ervah at the time of the brother's death. Once the brother died and left the tzarah to fall to yibum, she is no longer the tzarah of an ervah, and therefore she is permitted to undergo yibum.
As for the ailonis: an ailonis has no standing for the mitzvah of yibum, and she is not considered to be part of the framework of yibum at all. Consequently the tzarah is not considered the tzarah of someone who belongs to this framework, and therefore the tzarah is permitted.
The mishnah concludes by establishing that one cannot say about his mother-in-law, his mother-in-law's mother, and his father-in-law's mother that they were found to be an ailonis or that they performed mi'un. For by their very definition they had children, since that is how they became a mother-in-law, a mother-in-law's mother, and a father-in-law's mother, and therefore they cannot be an ailonis. So too regarding mi'un: rabbinic betrothal takes effect only with a minor girl who has not yet reached the age of mitzvos, and since they already had children, they are no longer minors. Therefore these cases are not relevant to them.
In summary: in this mishnah we learned the basic principle of the exemption of the 'tzarah of an ervah': a widow who is an ervah to the surviving brother is exempt from chalitzah and from yibum, and along with her, her tzarah and the tzarah of her tzarah are exempt until the end of the world. We listed the fifteen women to whom this law applies, and we clarified the cases in which the exemption falls away - she died, she performed mi'un, she was divorced, or she was found to be an ailonis - as well as the three women to whom these cases do not apply at all.