English Explanation of Mishnah Sanhedrin 1:3

The laying on of the elders’ hands and the breaking of the heifer’s neck [are decided upon] by three, according to Rabbi Shimon. But Rabbi Judah says: “By five.”
The rites of halitzah and “refusal” [are performed] before three.
The fruit of fourth year plantings and Second Tithes whose value is not known [are redeemed] before three.
Things dedicated to the Temple [are redeemed] before three.
Vows of evaluation to be redeemed with movable property, [are evaluated] before three. Rabbi Judah says: “One must be a priest.” [Vows of evaluation], [to be redeemed] with land [are evaluated] before nine and a priest.
And similarly [for the evaluation] of a man.
Mishnah three deals with the number of judges needed in cases that are of a religious/ritual nature.
Section one: According to Leviticus 4:13-21, when the whole community of Israel commits an accidental transgression, they must bring a bull as a sin offering. According to verse 15, before sacrificing the bull the elders would lay their hands on the bull’s head. Our mishnah teaches that this laying on of the hands was done by three judges.
The “breaking of the heifer’s neck” refers to Deut. 21:1-9. These verses describe a ritual of expiation that was to be done in the case where a person was found murdered but the murderer was unknown. Deuteronomy refers to “elders” who were to carry out the process and our mishnah teaches that there were three.
Section two: Halitzah is the refusal of the Levirate marriage (Deut. 25:5-10). If a woman’s husband dies and they have no offspring, his brother is obligated to marry her and bring forth offspring on his dead brother’s behalf. This is called “Levirate marriage”. If the brother should refuse to do so, they must go through a process called Halitzah before the woman is free to marry someone else. This is done in front of three judges.
“Refusal” refers to a daughter who was married off by her brother or mother. According to the Rabbis a father has a right to marry off his daughter while she is a minor and this marriage is totally binding and the girl cannot be released from the marriage except upon the death of the husband or divorce. However, a mother or brother’s ability to marry off the girl is less binding. When she becomes of a majority age she may refuse her husband and thereby annul the marriage. The “refusal” must be done in front of three judges.
Section three: Plants that are in their fourth year and the Second Tithe must be brought to Jerusalem and eaten there. If one lived far from Jerusalem and did not wish to carry all of this produce all the way to Jerusalem he could “redeem” the produce and bring the money to Jerusalem and use it to buy food there. The redeeming had to be done in front of three judges.
Section four: If a person dedicated an animal to the Temple that was not fit to be sacrificed, for instance a donkey, he could redeem the animal and donate the money to the Temple (Lev. 27:11). The redemption had to be done in front of three.
Section five: According to Lev. 27 a person could take a vow to donate his own value to the Temple. In such a case the Torah gives set amounts of money that must be donated to the Temple, depending on the age and gender of the one who took the vow. In general, since the Torah prescribes set amounts, no judges will be needed to evaluate how much the person owes. If however, the person has no money, he will need to donate some of his property. If the property to be donated is movable property a court of three is sufficient for its evaluation. According to Rabbi Judah, one of them must be a priest. If the property to be donated is land, a court of ten, including one priest is needed to determine the value of the land. According to Lev. 27:8, if the one who took the vow could not afford to donate his own value a priest was allowed to assess how much he could afford. According to the mishnah this assessment was done in front of a court of ten, which would include one priest.