English Explanation of Mishnah Sanhedrin 3:1

Cases concerning property [are decided] by three [judges].
This [litigant] chooses one and this [litigant] chooses one and then the two of them choose another, according to Rabbi Meir. But the Sages say: “The two judges choose the other judge.”
This [litigant] can invalidate this one’s judge, and this [litigant] can invalidate this one’s judge, according to Rabbi Meir. But the Sages say: “When is this so? When they bring proof against them that they are relatives or otherwise invalid; but if they are valid and experts, he cannot invalidate them.
This [litigant] may invalidate this one’s witnesses and this [litigant] may invalidate this one’s witnesses, according to Rabbi Meir. But the Sages say: “When is this so? When they bring proof against them that they are relatives or otherwise invalid; but if they are valid, he cannot invalidate them.
Chapter Three begins to discuss the court procedure in cases of financial matters, which only require three judges. The first mishnah discusses the selection of judges.
This mishnah contains three disputes between Rabbi Meir and the Sages with regards to the selection of judges and witnesses in cases concerning property disputes. All agree that the first two judges are selected by the litigants themselves, each litigant choosing one judge. However, Rabbi Meir and the Sages dispute with regards to the selection of the third judge. Rabbi Meir holds that the litigants together select a third judge and the Sages hold that the first two judges, those already selected by the litigants, are the ones to select the third judge.
With regards to the invalidation of the judges, Rabbi Meir holds that each litigant can indiscriminately invalidate the judge who was chosen by the opposing litigant. The Sages hold that the judges may only be invalidated on objective grounds, for either being relatives of the litigant or otherwise invalid. (We will learn more about the what cause a person to be invalid to be a a judge in mishnah three). If the judges are otherwise valid the opposing litigant may not disqualify them.
The Sages and Rabbi Meir have basically the same dispute with regard to witnesses. Note, that in this case Rabbi Meir’s opinion is much more radical. If a litigant can disqualify his rival’s witnesses without any due cause, how could anyone ever be convicted. The Talmud deliberates at length on this problem and makes several suggestions: 1) the litigant can only disqualify witnesses when there is only one witness. In such a case, since there are not the requisite number of witnesses, the litigant is not truly destroying his rival’s case; 2) the mishnah deals with a case where a person has two sets of witnesses, and the rival disqualifies only one set; 3) the rival has another witness who testifies with him that the other witnesses are disqualified; 4) the litigant claimed that the judges and witnesses were not valid. When it is established by independent evidence that he told the truth about the judges, he is believed with regard to the witnesses.
In any case, from the fact that there are four solutions to this problem, we can see how puzzling Rabbi Meir’s opinion truly is.