We are studying the second Mishnah in the eleventh chapter of Maseches Yevamos. The Mishnah deals with a son whose father is unknown, and with the consequences of that uncertainty for the obligation of yibum and chalitzah between him and his brothers.
The case in the Mishnah:
"Mi shelo shahasah achar ba'alah sheloshah chodashim, venisses veyaldah" - a woman whose husband died, whom the halachah requires to wait three months before marrying another man, and she did not wait, but married and gave birth. The birth took place seven months after her second marriage, "ve'ein yadua im ben tishah larishon im ben shivah la'acharon" - and we have no way to determine whether this is a nine month child from the first husband who died, or a seven month child from the second husband. This is precisely the reason she was required to wait three months, and since she did not wait, the uncertainty arose as to who the child's father is.
Sons from the first husband and sons from the second - brothers through the mother:
"Hayu lah banim min harishon uvanim min hasheni" - this woman has other sons from the first husband and other sons from the second husband, besides that son whose status is in doubt. It turns out that they are all certainly brothers to one another through the mother, since they all share one mother, while as for the father, it is unknown whether they share the same father or not.
If the son of doubtful status grew up, married a woman, and died childless, his widow comes before those brothers for yibum. If they are brothers only through the mother, his widow is forbidden to them as an ervah - the wife of a brother who was not alive in his lifetime; but if they share the same father, the obligation of yibum applies to them.
Therefore we learn "choltzin velo meyabmin" - they perform chalitzah, because there may well be an obligation of yibum here; and they do not perform yibum, because the widow may be an ervah to them, if they are his brothers only through the mother.
"Vechein hu lahem" - and the same law applies in reverse: if one of those brothers died and the son of doubtful status is the survivor, he performs chalitzah for the widow and does not perform yibum, since it is possible that he is his brother only through the mother and not through the father.
Brothers who are not from the same mother:
"Hayu lo achim min harishon ve'achim min hasheni shelo me'osah ha'em" - the son of doubtful status has possible brothers from the first husband and possible brothers from the second husband, except that they are not sons of the same mother. Here the picture is reversed: there is no concern at all of a brother's wife through the mother, since they certainly have no mother in common. It is possible that they are his brothers through the father and the obligation of yibum applies, and it is possible that there is no relationship between them at all.
Therefore, if one of the brothers died and his widow came before him, the son of doubtful status may perform chalitzah or yibum - he has the option of performing chalitzah and he has the option of performing yibum. For if he is his brother through the father, there is an obligation of yibum here; and if they have no common parent at all, there is no ervah here, and yibum is permitted.
And if it is the son of doubtful status who died, and his widow comes before those brothers - who may share a father with him, but certainly do not share a mother with him - then "echad choletz ve'echad meyabem" - one performs chalitzah and one performs yibum. The reason is this: it is possible that one of the two is the true yavam and the other is not, and yibum should not be performed first, since if she is not his yevamah at all, she needs chalitzah in order to be permitted to marry an outsider. Therefore one of the brothers performs chalitzah first, and afterwards the second performs yibum.
In summary: In this Mishnah we learned the law of a son born to a woman who did not wait three months after her husband's death, where it is unknown whether he is a nine month child from the first husband or a seven month child from the second. When the brothers are sons of the same mother - they perform chalitzah and not yibum, because of the possible obligation of yibum on the one hand and the possible ervah on the other, and the same applies in reverse. And when the brothers are not sons of the same mother - he may perform either chalitzah or yibum, while as for them, one performs chalitzah and one performs yibum.