We are in tractate Nedarim, chapter 3, Mishnah 2. This Mishnah is a bit complex, similar to the previous Mishnah - not because the topic itself is difficult, but because it is hard to see how the Gemara understands the Mishnah from its actual wording. Nevertheless, we will try to explain it to the best of our ability.
The Starting Point:
Although a valid court, containing valid and ordained judges, is generally required, the truth is that two litigants who disagree regarding a monetary debt may resolve their doubt themselves. They can flip a coin, or even say: 'I do not mind, I will pay you in any case'. The one giving the money may give it of his own volition unilaterally, and if it can be resolved by flipping a coin - it can certainly be resolved by a judge who is not ordained or is invalid. Therefore, the litigants may agree between themselves: 'Let us go to my father, who is a wise man, and if he rules that there is no debt here - I will waive my rights and forgo it'. Such an agreement is binding, even though the father is invalid to serve as a judge due to his relation.
The question under discussion is: when the two parties have accepted upon themselves through a binding agreement the ruling of an invalid judge or judges - are they allowed to retract this retroactively, after the verdict? Rabbi Meir and the Sages disagree on this:
Rabbi Meir: He can retract his consent to pay, since the burden of proof rests on the one who seeks to extract money from his fellow, and the defendant has the ability to say: 'I do not want to pay; as for my father saying that I must pay - I will not listen to him'.
The Sages: Once he has accepted upon himself a binding commitment to pay, and after the Torah litigation has concluded and ruled against him, he is obligated to uphold his commitment.
The wording of the Mishnah:
"Amar lo: ne'eman alai abba, ne'eman alai avicha" - one of the parties says that his own father or his friend's father will decide between them, and they both agree. Even though a person's father is a relative and is invalid to judge, since they agreed to accept him - it is effective, as explained above.
"O ne'eman alai sheloshah ro'ei bakar" - they agreed to accept the decision of three cattle herders, simple cowboys, to serve as judges.
The reason the Mishnah specifically mentioned cattle herders, and not just three ordinary laymen, is that cattle herders are not only lacking in halachic expertise: they are not even well-versed in worldly affairs, they may never have seen a contract in their lives, and do not necessarily know even basic commercial and social norms. The novelty is that even these people, if the parties agreed to appoint them as their arbitration panel - their agreement is a valid agreement.
"Rabbi Meir omer: yachol lachzor bo" - even after the invalid judges have ruled, the one whom the ruling went against is permitted to say: 'I have changed my mind, we will not listen to them, since they are not a valid court'.
"Vachachamim omerim: eino yachol lachzor bo" - once he has agreed to accept their ruling, there is no practical difference that they are invalid by the letter of the law; since he committed to be bound by their words, he is bound by them and cannot retract this retroactively. The Sages concede that prior to the verdict, while they are still in the middle of the Torah litigation, the litigant may say, 'Your father is biased', or, 'These cattle herders are more foolish than I assumed', and withdraw to turn to a regular court. However, once the verdict has been issued and the trial is concluded - he can no longer retract it.
The second part of the Mishnah - accepting an oath:
The second part of the Mishnah deals with a related matter: administering an oath in court. Under normal circumstances, a Torah oath is taken in order to become exempt from payment: the person was supposed to pay, and the law is that if he swears - he will be exempt. There are three cases in which a person is obligated to take a Torah oath. One of them is a single witness: Reuven claims that Shimon owes him money, and he only has one witness; Shimon denies it. We do not obligate Shimon to pay based on one witness, but by Torah law we obligate him to swear that he does not owe the money, and thereby he will be exempt from the payment.
How is this oath administered? A Torah oath involves holding an object, and the classic object is a Sefer Torah. The one taking the oath stands in beis din and holds the Sefer Torah, swearing by the name of Hashem or one of His designations, saying: "Just as I believe in Hashem, so too I believe that I do not owe you the money, and I hereby swear." Such an oath is valid and exempts him from payment.
The question the Mishnah seeks to clarify is the law when the oath is taken in order to extract money. Generally, one does not swear to extract money, but there are exceptions. As we saw in Tractate Bava Basra, a day laborer who claims within the time the employer is supposed to pay him, "I did not receive my wages," while the employer claims, "I already paid you" - beis din believes the laborer. The reason: the employer is busy with many matters, whereas the worker stands and waits for his wages with only one thing on his mind, so he is not mistaken. Therefore, if the laborer swears immediately at the time the money is due to him that he still has not received his wages - we believe him and extract money based on his oath. This is highly exceptional, and this is the case our Mishnah addresses.
We find an employer and a worker in beis din: the worker claims he did not receive his wages, and the employer claims he paid. The worker must swear in order to collect his wages. However, holding an object - grasping a Sefer Torah, swearing, and mentioning the name of Heaven - is a weighty and intimidating matter, and a person seeks to avoid it. Therefore, the employer says to him: 'You do not need to take this severe oath, a light oath is enough for me - swear to me by your life, and there is no need to mention the name of Heaven.' The worker then swears: 'I hereby swear by my life that you did not pay me.'
Is this sufficient, or may the employer say after the fact: 'Although I said an oath by your life is enough for me, now that I have seen this - it is not enough for me, and I request a severe oath on a Sefer Torah'? That is: can the oath be rejected after it has already been taken? This is the exact same principle. Rabbi Meir holds that he has the right to do so, for he can say: 'I will not pay the money until I receive what I ask for, and I must be satisfied with the process.' And the Sages say: Once he agreed, he obligated himself, and he cannot retract it.
"Hayah chayav lachaveiro shevuah, ve'amar lo: dor li bechayei roshecha" - a person who was obligated to swear to his fellow, and again, this is a case where money is extracted based on an oath, such as a worker swearing to his employer that he did not receive his wages. The word "dor," from the root of neder (vow), refers to an oath: the employer says to him, 'Swear to me by the life of your head, that is enough for me, and there is no need for a Sefer Torah and the name of Heaven.' The worker swears: 'I hereby swear by my life that you did not pay me,' and now the employer retracts and says: 'This is not enough for me, and I do not want to pay.' Is this effective?
"Rabbi Meir omer: yachol lachzor bo" - the employer may say: 'I changed my mind, take out a Sefer Torah and swear by the name of Hashem.'
"Vachachamim omerim: eino yachol lachzor bo" - once he agreed to accept this light oath, an oath by the life of his head, he can no longer retract it; he is bound by his agreement and must pay. And the halachah follows the opinion of the Sages.
It is worth noting that in the opposite case, everyone agrees: when the oath comes to exempt a person from payment, and the opposing party says, 'Just swear to me and I will believe you, and I will forgo my claim' - everyone agrees that once he swears, even by the life of his head, if that was agreed upon, the claim is forgiven and pardoned, and he cannot sue again. The question in our Mishnah applies specifically to a case where they are extracting money: Rabbi Meir holds that we do not compel a person to give money unless the Torah obligated him to do so, and here there was no Torah obligation; whereas the Sages hold that once he agreed, we compel him to pay even in such a scenario.