Yom Kippur has two Torah readings, during Shachrit in the morning and during Mincha in the afternoon.
Morning (Shachrit)
The morning Torah reading is from Leviticus 16:1-34. It is read during Yom Kippur because it recounts the procedure for the Kohen Gadol (High Priest) to make atonement for the community of Israel (“Klal Yisrael”). It follows, in the Torah, after the deaths of Nadad and Abihu, Aaron’s two oldest sons, who entered the Holy of Holies (the innermost sanctum of the Tabernacle) inappropriately.
The chapter opens with Hashem warning Aaron not to come in to the Holy of Holies “at all times”. He is only to enter with a bull for a sin-offering and after he has clothed himself in sacred white linen garments after he has purified himself in water. He first atones for himself and his family.
Aaron is then to take two goats and choose by lot which goat will be sacrificed as atonement/expiation for Israel and which goat would be sent into the wilderness for Azazael, which means “strong and mighty”, for Hashem.
Aaron is then to take incense and burn it so that a cloud of smoke covers the inner sanctum. He is then to sprinkle blood from the goat sacrifice seven times on the altar. He is then to go to outer altar (where daily sacrifices are held) and sprinkle blood from both the bull and the goat. This makes atonement for himself, his family and Israel.
In sending the other goat out into the Wilderness, he is to first lean his hands on its head and confess all the sins, willful and unwilful. He then sends the goat out accompanied by a designated man.
Finally, this is commanded as an eternal decree. In the seventh month, on the tenth of the month, we are to afflict ourselves and no one in our community is to do any work. It is a Shabbat of complete rest.
Afternoon (Mincha)
The afternoon Torah reading is Leviticus, chapter 18. Hashem tells Moses that when they come into the land, they are not to do the practices of the Egyptians from whence they came, nor the practices of Canaanites to where they’re going. The portion then lists various forbidden relationships, such as man with his sister or father’s wife. They are also told not to pass their children through fire.