We are in the fourth chapter of Tractate Sanhedrin, Mishnah 2. The mishnah opens: "Dinei hatumaos vehataharos" - laws of tumah and purity, meaning laws that are not capital cases. Even though this discusses tumah and purity, a type of case in which there is no winning party and losing party and therefore there was reason to separate it from other cases, the version before us in the Babylonian Talmud is "Dinei mamonos, hatumaos vehataharos" - laws of monetary cases, tumah, and purity, including monetary cases as well. All of these are juxtaposed here as a contrast to capital cases.
The order of expressing opinions in the beis din:
In monetary cases, tumah, and purity - "Mathilin min hagadol" - they begin from the greatest: when the judges come to express their opinions, they start with the senior member of the beis din, in order to accord him honor. While he is permitted to waive his honor, the proper procedure initially is to hear the greatest one first.
In capital cases - "Mathilin min hatzad" - they begin from the side: they start specifically with the junior judges, meaning the twenty-third judge in rank, and proceed upwards from there.
The reason for this: We do not want a judge to be afraid to express his opinion. If the greatest of the generation were to open and say "guilty", the other judges would hesitate to disagree with him. Because of this, we start with the junior judge, so that everyone will feel comfortable stating their true and honest opinion.
Why is this called "min hatzad" - from the side?
As we will see in the next mishnah, when the Sanhedrin convened - whether as a court of twenty-three or seventy-one - it sat in the shape of a semi-circular threshing floor. In the middle of the semi-circle sat the most prominent judge, to his one side sat number two, and to his other side sat number three. It turns out that the senior judges sat in the center and the lower-ranking ones sat at the edges, and the furthest away were the last of the twenty-three or the seventy-one. Hence the expression "from the side".
Eligibility of the judges:
The mishnah continues: "Hakol kesherin ladun dinei mamonos" - everyone is eligible to judge monetary cases. Strictly speaking, semichah is required, as was mentioned earlier, but in terms of personal eligibility, the matter is determined solely by qualifications: anyone who is fit to judge because of his expertise may serve as a judge.
By contrast: "Ve'ein hakol kesherin ladun dinei nefashos ela kohanim, levi'im veyisre'elim hamasi'in lakehunah" - but not everyone is eligible to judge capital cases, except kohanim, levi'im, and yisraelim who can marry off their daughters into the kehunah. In capital cases, not just any judge is accepted, no matter how great a Torah scholar he may be, rather he must be of pure lineage. Whether a kohen, a levi, or a yisrael - provided that he is of such pure lineage that his daughters are eligible to marry into the kehunah.
Nesinim and mamzerim - are not eligible to serve as judges in capital cases. However, a mamzer is eligible to judge monetary cases, since "hakol kesherin" - everyone is eligible, in contrast to capital cases.
Gerim - A ger himself is disqualified for capital cases, and so is someone whose parents are both gerim. For someone who has no Jewish blood from his parents, it is derived from the verse that he cannot hold a position of authority, such as a king, and therefore he cannot serve as a judge. However, someone whose one parent is a ger and the other is not a ger is eligible. As long as there is a Jew by birth in the direct family tree and the person is his descendant, he is eligible to judge monetary cases; whereas for capital cases this is not sufficient, since a giyores cannot marry a kohen.
In summary: In this section we learned the distinction between monetary cases, tumah, and purity, and capital cases in two matters: in the order of expressing opinions, "mathilin min hagadol" - they begin from the greatest, versus "mathilin min hatzad" - they begin from the side, so that the honest opinion of the junior judge will not be withheld; and in the eligibility of the judges, "hakol kesherin ladun dinei mamonos" - everyone is eligible to judge monetary cases, whereas in capital cases a lineage of "hamasi'in lakehunah" - who can marry off their daughters into the kehunah, is required.