Shemini
Leviticus 9:1-11:47

On the eighth day following "seven days of 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."

In preparation for the upcoming festival of Passover, when every Jew had to be in a state of ritual purity, the section of Parah (Numbers 19) is added to the weekly reading this week. Parah relates the laws of the Red Heifer with which a person contaminated by contact with a dead body was purified.

Tzav
Leviticus 6:1-8:36

G-d instructs Moses to command Aaron and his sons regarding their duties and rights as Kohanim ("priests") who offer the korbanot (animal and meal offerings) in the Sanctuary.

The fire on the Altar must be kept burning at all times. In it are burned the wholly consumed Ascending Offering; veins of fat from the Peace, Sin, and Guilt Offerings; and the "handful" separated from the Meal Offering.

The Kohanim eat the meat of the Sin and Guilt Offerings and the remainder of the Meal Offering. The Peace Offering is eaten by the one who brought it, except for specified portions given to the Kohen. The holy meat of the offerings must be eaten by ritually pure persons, in their designated holy place, and within their specified time.

Aaron and his sons remain within the Sanctuary compound for seven days, during which Moses initiates them into the priesthood.