Leviticus 9:1–11:47
The name of the Parshah, "Shemini," means "eighth" and it is found in Leviticus 9:1.
On the eighth day, following the seven days of their inauguration, Aaron and his sons begin to officiate as kohanim (priests); a fire issues forth from G‑d to consume the offerings on the altar, and the divine presence comes to dwell in the Sanctuary.
Aaron's two elder sons, Nadav and Avihu, offer a "strange fire before G‑d, which He commanded them not" and die before G‑d. Aaron is silent in face of his tragedy. Moses and Aaron subsequently disagree as to a point of law regarding the offerings, but Moses concedes to Aaron that Aaron is in the right.
G‑d commands the kosher laws, identifying the animal species permissible and forbidden for consumption. Land animals may be eaten only if they have split hooves and also chew their cud; fish must have fins and scales; a list of non-kosher birds is given, and a list of kosher insects (four types of locusts).
Also in Shemini are some of the laws of ritual purity, including the purifying power of the mikvah (a pool of water meeting specified qualifications) and the wellspring. Thus the people of Israel are enjoined to "differentiate between the impure and the pure."
HACHODESH IN A NUTSHELL
Exodus 12:1–20
This being the Shabbat that falls on or before the first of Nissan, we also read the section of Hachodesh (Exodus 12:1–20), which relates G‑d's words to Moses in Egypt two weeks before the Exodus, instructing us to set the Jewish calendar by the monthly new moon, and to regard Nissan as the "head of months." G‑d also instructs to bring the Passover offering, to eat it with matzah and bitter herbs, and to abstain from leaven for seven days.
Ezekiel 45:18-46:15 (Hachodesh)
This special haftorah is a prophecy regarding the Paschal Offering that will be brought during the Messianic Era, reflecting the theme of the Hachodesh Torah reading—Moses' command to the Israelites in Egypt to prepare and bring the Paschal lamb.
This haftorah is part of Ezekiel's prophecy regarding the third Holy Temple—its structure, inauguration and some of the practices that will be observed therein.
The haftorah begins with a description of the various sacrifices that will be offered during the Temple's seven-day inauguration ceremony, and then mentions that on the 14th of Nissan we shall bring the Paschal offering.
Much of the rest of the haftorah is devoted to the sacrifices that will be brought by the "leader," and prescribes his entry and exit from the Temple.
This chodesh (new moon, month) shall be for you the head of months; it shall be for you the first of the months of the year (12:2)
G‑d showed Moses the new moon at its moment of rebirth, and said to him: "When the moon is reborn, mark the beginning of a new month."
(Mechilta)
There was a large courtyard in Jerusalem called Beth Yaazek, where all the witnesses (who had seen the appearance of the new moon) used to assemble, and the beit din (rabbinical court) used to examine them. They used to entertain them lavishly there, so that they should have an inducement to come . . .
The pair of witnesses who arrived first were crossexamined first. The senior of them was brought in and they said to him: "Tell us how you saw the moon—in front of the sun or behind the sun? To the north of it or the south? How big was it, and in which direction was it inclined? How broad was it?" . . . Rabban Gamaliel used to have diagrams of the phases of the moon on a tablet on the wall of his upper chamber, and he used to show them to the unlearned and ask, "Did it look like this or this?" . . .
After that they would bring in the second witness and question him. If their accounts tallied, their evidence was accepted. The other pairs were questioned briefly—not because they were required at all, but so that they should not be disappointed and discouraged from coming (the next time).
The head of the beit din would then proclaim: "Sanctified!" and all the people would repeat after him, "Sanctified! Sanctified!"
(Talmud, Rosh Hashanah ch. 2)
(Shemini) https://w2.chabad.org/media/pdf/14405.pdf
