In the previous mishnah we learned that if a man tells his wife she is permitted to any man except for one particular individual, the Sages ruled that this is not a valid get. Our mishnah explains this ruling and draws a distinction between cases, and everything depends on who that individual is.
The valid exception - someone with whom kiddushin cannot take effect:
The mishnah states: if a man tells his wife she is permitted to any man, "chutz me'avi ume'avich, me'achi ume'achich, me'eved umigoy" - except for my father and your father, my brother and your brother, a slave and a non-Jew - the get is valid.
The principle that unites all of these people is: "kol she'ein bo tefisas kiddushin" - anyone with whom kiddushin cannot take hold. These are people with whom it is halachically impossible to effect a valid betrothal. It is not merely that such a marriage is forbidden; the kiddushin simply do not take effect at all.
Therefore the get is valid: these people are not really an exception to the get's effectiveness at all, since there was never any connection to them in the first place - not before the marriage, not during it, and not after it. They are entirely irrelevant, and so excluding them is not considered a genuine exception.
The exception that invalidates - someone with whom kiddushin can take effect:
The mishnah continues: if a man tells his wife she is permitted to any man "chutz me'almanah lekohen gadol, gerushah vachalutzah lekohen hedyot, mamzeres unesinah leYisrael, bas Yisrael lemamzer velenasin" - except for a widow to a Kohen Gadol, a divorcee or chalutzah to an ordinary kohen, a mamzeres or nesinah to an ordinary Jew, and a Jewish woman to a mamzer or a nasin - the get is invalid. These are the cases:
Almanah lekohen gadol - a widow who marries a Kohen Gadol.
Gerushah vachalutzah lekohen hedyot - a divorcee, or a woman who underwent chalitzah, who marries an ordinary kohen.
Mamzeres unesinah leYisrael - a mamzeres or a nesinah who marries an ordinary Jew.
Bas Yisrael lemamzer velenasin - a Jewish woman of unblemished lineage who marries a mamzer or a nasin.
What all of these have in common: these are people with whom kiddushin does take effect, even though the marriage itself is forbidden.
And why is the get invalid? Because even though she is not permitted to marry these men, kiddushin with them does take hold - if she were to marry one of them, his betrothal would be halachically valid. Therefore, when the husband tells her that the get does not permit a marriage between her and one of these men, this invalidates the get, since there is no complete severance here. There is no full separation between him and her, for even after the get his connection to her remains in that she is not permitted to these men. And even though marrying such men is a transgression, since the kiddushin would take effect were she to marry one of them - even in violation of the law - the get is invalid.
In summary: The mishnah distinguishes between two types of exceptions. If a man excludes from his wife's permission people with whom kiddushin cannot take effect - his father and her father, his brother and her brother, a slave and a non-Jew - the get is valid, since these men were never relevant to begin with. But if he excludes people with whom kiddushin does take effect, even though marriage to them is forbidden, the get is invalid, because it lacks complete severance.