The First Case - "Meis Ba'alech V'achar Kach Meis B'nech":
A woman whose husband and son went overseas, and messengers came and told her that her husband died and afterward her son died. According to this report the son was alive at the time of the father's death, so the husband did not die childless; the woman is not subject to yibum, and she is permitted to marry anyone she wishes. On the strength of this report she married another man, without any need to be concerned about yibum at all.
Afterward they told her that matters were the reverse: the son died first and the husband died after him, so the husband did die childless and she was subject to yibum, and she had no right to marry outside the family. Therefore the Mishnah says "teitzei" - she must leave the marriage she entered, since a yevamah who marries an outsider transgresses a prohibition, and she must now receive chalitzah or undergo yibum.
"V'havalad Rishon V'acharon Mamzer":
The first child - the child born to her while she assumed that her first husband had died, when in truth he was still alive and died only afterward. This child is a mamzer, since she was a married woman.
The last child - the child born to her from the outsider she married, after the husband's death, when she was already subject to yibum and was a yevamah who married an outsider. This child is a mamzer only according to Rabbi Akiva, who holds that even in cases of ordinary prohibitions that carry no kareis the child is a mamzer. It emerges that our Mishnah follows the opinion of Rabbi Akiva, who rules that even the last child, born after the husband's death through the unwitting transgression of a yevamah marrying an outsider, is a mamzer.
The Second Case - "Meis B'nech V'achar Kach Meis Ba'alech":
They told her that her son died and afterward her husband died. According to this report the husband died childless and she is subject to yibum, and on that basis she underwent yibum with her husband's brother. Afterward they told her: "chiluf hayu hadevarim" - the events occurred in the opposite order, so she was never obligated in yibum at all, and the yibum that was actually performed is a forbidden marriage to a brother's wife, which carries kareis.
Therefore "teitzei": the marriage is invalid and is no marriage at all, and there is not even a betrothal here, since their intention was for yibum.
Here too the Mishnah rules that "havalad rishon v'acharon mamzer" - the first child and the last child are both mamzerim:
The first child - born while the husband was still alive, through the prohibition of a married woman, which is more severe than the prohibitions carrying kareis and falls under those punishable by death at the hands of the court.
The last child - born from the yavam after the husband's death, through the prohibition of a brother's wife, which carries kareis. For that reason he is a mamzer even according to the Sages who disagree with Rabbi Akiva.
The Third Case - "Meis Ba'alech" and Afterward "Kayam Hayah U'meis":
This case has nothing to do with yibum at all, and it may well be speaking of a couple who have children. They told her "meis ba'alech" - your husband died - and on that basis she married, and afterward they told her "kayam hayah u'meis" - at the time she married he was still alive, and he died only afterward. "Teitzei" - she must leave the second husband. Even though she is now no longer married to the first husband, since he ultimately died, nevertheless at the time of the marriage she was married to a living man, so that marriage was no marriage at all; and from now on she is forbidden to the second man, since she is one who had relations with him in a forbidden manner.
"V'havalad harishon mamzer" - the child born to her while the first husband was still alive is a mamzer.
"V'ha'acharon" - the child born to her after the first husband's death is not a mamzer, even though she must leave the second man, since the first husband is no longer alive.
Where She Was Only Betrothed:
They told her that her husband died and she was only betrothed, without entering the chuppah, and afterward her husband returned - she is permitted to return to him, since there was no forbidden relationship here at all. More than that: even if the second man, who betrothed her, gave her a get, that get takes no effect as a get, and it does not even disqualify her from marrying a kohen as a divorcee, since there is no divorce here at all, for she was never married to that man.
On this point a derivation is stated from the wording of the verse that forbids a kohen to marry a divorcee: "ishah gerushah me'ishah" - a woman divorced from her husband, and not from one who is not her husband. Therefore when a man who is not her husband, as in our case, gives her a get, that get has no substance, and she is not considered a divorcee, and she remains permitted to a kohen.
In summary: we have learned four situations of contradictory testimony. A woman who married an outsider on the strength of a report that the son was alive at the time of the husband's death, and it turned out that the husband died childless - she must leave, and both her first child and her last child are mamzerim according to Rabbi Akiva. A woman who underwent yibum on the strength of a report that her son died first, and it turned out that the events occurred in the opposite order - she must leave, and her first child and her last child are mamzerim even according to the Sages, since we are dealing with prohibitions carrying kareis and with prohibitions punishable by death at the hands of the court. A woman who married on the strength of a report of her husband's death, and it turned out that he was still alive and died only afterward - she must leave, and only her first child is a mamzer. And a woman who was only betrothed - she returns to her husband, and the get of the second man does not disqualify her from marrying into the priesthood.