Yevamos, Chapter Ten, Mishnah 2. This mishnah is a direct continuation of the previous one, and it draws the distinction between a woman who remarried on the strength of the testimony of a single witness and one who remarried on the strength of two witnesses, and how that distinction affects her obligation to bring a sin offering for having lived with a man forbidden to her.
The two cases in the mishnah:
"Nises al pi beis din" - she married on the ruling of beis din, meaning that she heard the testimony from a single witness only, and beis din applied their exceptional enactment and permitted her to marry. If the first husband then appears - "teitzei" - she must leave, as was explained in the previous mishnah, and she cannot remain married even to her first husband. However, "peturah min hakorban" - she is exempt from the offering, since she relied on the ruling of beis din.
"Lo nises al pi beis din" - she did not marry on the ruling of beis din, but rather on the strength of two valid witnesses. Here too "teitzei" - she must leave, and she cannot remain married to him, but "chayeves bakorban" - she is obligated to bring an offering, since she is a full-fledged unintentional sinner. Even though the act was done unintentionally, the responsibility rests on her, for she has no ruling of beis din to lean on.
"Yafeh koach beis din":
The mishnah sums up: "Yafeh koach beis din" - the power of beis din is fine and strong, for it exempts a person from the offering. Once the woman relied on their ruling, she is no longer obligated to bring an offering.
However, the halachah does not follow this mishnah. The ruling is that even a woman who married on the instruction of beis din, she and her second husband are both obligated to bring an offering. The Gemara explains the reason: this is not a ruling and not a halachic decision at all, but simply a mistake.
"Horuha beis din linasei vehalchah vekilkelah":
Beis din ruled that she is permitted to marry, and according to our mishnah that alone is enough to exempt her from the offering. But "shehalchah vekilkelah" - she went and ruined it: instead of marrying someone permitted to her, she married someone forbidden to her, such as where her husband is a kohen and she is a divorcee, or the like. Once she did something improper with that ruling - "chayeves bakorban" - she is obligated to bring an offering for having lived as a wife with this forbidden husband.
And the reason is: "shelo hitiru lah ela linasei" - beis din permitted her only what was proper, to marry a man permitted to her. Once she married someone forbidden to her, she is no longer considered as one who is following the ruling of beis din, but as one acting on her own judgment. Even though the very permission to marry, based on the testimony about her husband's death, rests on beis din, she did not rely on them fully, since she went on and did something improper with that ruling - and therefore she is obligated to bring an offering.
In summary: This mishnah distinguishes between a woman who married on the ruling of beis din, who must leave but is exempt from the offering, and a woman who married on the strength of two witnesses alone, who must leave and is obligated to bring an offering. In this sense "the power of beis din is fine," for it exempts from the offering, although the halachah is that even in this case she and her second husband are obligated to bring an offering, since this is not a ruling but a mistake. And if beis din instructed her to marry and she went and ruined it, she is obligated to bring an offering, for they permitted her only to marry properly.