In last week's Parshah we spoke about the seven days in which Moses and Aaron inaugurated the Mishkan. In this Parshah, we learn about the following day, which is what Shemini means—the eighth.
Aaron and his sons officially begin their jobs as Kohanim—priests serving in the Mishkan. First they prepare a sacrifice, and then all the Jews come to the entranceway of the Mishkan to watch as they bring the sacrifice up on the altar. Then, Moses and Aaron bless the people and everyone sees as a fire came down from heaven and consume the sacrifice. The people are so overwhelmed—and overjoyed—by this amazing sight that everyone begins to sing praises to G‑d.
But then, in middle of all the celebration and excitement, something terrible happens. Two of Aaron's sons, Nadav and Avihu, light a fire to bring a sacrifice that they aren't supposed to bring, and as punishment, they die. You can imagine how sad this tragedy makes Aaron, but he does not say anything, he remains quiet and accepts G‑d's judgment. Moses tells Aaron and his remaining sons, Eleazar and Itamar, to continue doing the service in the Mishkan, and they do, while the rest of the People of Israel mourn the death of Nadav and Avihu.
Next we learn about a law that G‑d tells Aaron: Whenever they will be serving in the Mishkan, Aaron and his sons, the Kohanim, are not allowed to drink wine and become drunk.
G‑d tells Moses to tell them about the animals that may be eaten and those that may not be eaten. Kosher animals—those that are permissible—have two signs: they chew their cud and have split hooves. Kosher fish must have fins and scales. We are given a list of the non-kosher birds, and a list of four types of locusts which are kosher; the Torah tells us that all other insects are not kosher.
We also learn about the idea of purity, which is like a spiritual type of cleanliness. If something becomes impure, by touching something impure, for example, the dead body of a non-kosher animal, then it must be immersed in a special body of water called a mikvah. The mikvah has a special power to restore purity to things (like vessels and dishes) and people that have become impure.