Sefer Vayikra consists primarily of commandments, with a bit of narrative here and there. Some of the commandments were taught in the Mishkan, and some were taught on Mt. Sinai.
Vayikra starts with God calling to Moshe from the Mishkan (Vayikra 1:1), switches back (in time and place) to Mt. Sinai (7:38, see Rashi on 8:2), switches back to the Mishkan in our Parasha (10:8, see Gittin 60b), and then goes back again to Mt. Sinai at the end of the Sefer (25:1, 26:46 and 27:34).
Part of the “back and forth” between the two locations is based on the Torah describing the same day—the day the Mishkan becomes fully operational—twice. The first time, at the end of Sefer Shemos, was the culmination of the Mishkan’s construction, which led to the communication that opens Sefer Vayikra. The second time, in our parsha, פרשת שמיני, is the culmination of the שבעת ימי המילואים (The Seven Days of Initiation), see Rashi on 9:1.. The formula is fairly straightforward: before the Mishkan was fully operational, God spoke to Moshe on Mt. Sinai; after it became fully operational, He spoke to him in the Mishkan.
R’ Levi tells us that “eight sections were taught on the day the Mishkan became fully operational” (Gittin 60a). Since the transition from God communicating with Moshe on Mt. Sinai to communicating with him in the Mishkan occurred on that day, we might have expected those eight sections to be taught in our parsha. However, according to Rashi (and Tosfos), only one of them is—the prohibition against drinking wine before serving in the Mishkan (10:8-11). Ran and Rabbeinu Crescas (Gittin 60a) suggest that “Parshas Temayim” (another of the eight sections) refers to the “tumah” (impurity) caused by dead animals (11:29-43)—which is taught at the end of our Parsha—and continues through the “tumos” caused by childbirth, “tzara’as,” and bodily emissions (12:1-15:32). This would mean that two of the eight sections were taught in our parsha (the second one continuing over the next two parshiyos as well), with the others taught later in Sefer Vayikra and in Sefer Bamidbar.
Sifre Zuta (7:11, see also Mishnas Rebbe Eliezer 6) says that God spoke to Moshe 15 times that day—listing the 15 communications—with only one of them appearing in our parsha (the prohibition against drinking wine before serving in the Mishkan). Besides trying to understand why there are two different lists, two of R’ Levi’s eight aren’t included in Sifre Zuta’s list, giving us a total of 17 communications.
Going through the two lists to try to explain their differences is beyond the scope of this piece (although I did make an attempt in 5776). Nevertheless, from the context of the Talmud (in Gittin) it is apparent that R’ Levi’s list is based on which sections were not only communicated that day, but also written down then, while Sifre Zuta’s list includes communications that weren’t written down right away. There must be something about the eight sections on R’ Levi’s list that made it necessary to write them down right after they were taught, so that they could be referred to—in writing—soon after the Mishkan was built.
When was the rest of the Torah written down? The Talmud (in Gittin) discusses whether the Torah was written down “piece by piece,” as Moshe was commanded each section, or was first written down—in its entirety—shortly before he passed away, after everything had been commanded. [Actually, that’s how Rashi understands the discussion; Tosfos understands it to be whether each section was written down as it was commanded or in the order it appears in the Torah. The difference between these two approaches doesn’t really impact the issue I am about to bring up.] R’ Levi’s statement is used to explain why, according to the opinion that the Torah wasn’t written down until everything was commanded, it’s referred to as a “מגיל,” which implies it was written down piece by piece rather than all at once. Since these eight sections were written down on the “eighth day,” they are the exception, and when the rest of the Torah was written down over 38 years later, the “pieces” before, in-between and after these eight were written down. To summarize, with the exception of the eight sections R’ Levi mentions, the Torah wasn’t written down until all of it was commanded (at least according to this opinion).
However, according to R’ Akiva (Zevachim 115b), all of the Mitzvos, including their details, were taught on Mt. Sinai, and repeated in the Mishkan. [Rashi, in his commentary on Vayikra 25:1, seems to follow R’s Akiva’s opinion.] If everything had already been commanded on Mt. Sinai, why couldn’t the Torah have been written down sooner, rather than waiting until everything was taught a second time in the Mishkan?
Those of us raised “frum from birth” learned each aspect of religious life as we experienced it growing up. Those who left Egypt didn’t have this advantage, so they were taught things “piece by piece.” Some things were taught before they left Egypt (e.g. circumcision and Pesach); some were taught before they reached Mt. Sinai (e.g. the Sabbath and the civil laws); some were taught during their stay at Mt. Sinai; and some were taught after they left Mt. Sinai. Even if everything was taught to Moshe on Mt. Sinai, nothing was shared with the nation until God told him it was time to share it. (According to some, this is why each communication with Moshe opens with “לאמר,” as Moshe was now being given permission to share it with everyone else.) Ever wonder why Moshe needed to ask God about some laws (e.g. how to deal with the blasphemer, or the inheritance of the daughters of Tzelafchad) if he had already been taught everything at Sinai? Although numerous answers are given, one answer could be that even though Moshe knew the answer, he needed to ask God if the time to share it was now. Similarly, even if Moshe had already been commanded everything at Sinai, he couldn’t write it down until it was time to share it with the nation. And according to the opinion that the Torah had to be fully complete when it was written down, it couldn’t be written down until he was given permission to share it all.
Rabbi Dov Kramer wrote a weekly dvar Torah from 5764-5776. His comparison between the two lists of what was taught on the “Eighth Day” is available at https://rabbidmk.wordpress.com/2016/03/31/parashas-shemini-5776/.