This parsha continues after parsha Shmini of two weeks ago where Aharon's two sons passed away when they offered an unauthorized offering in the Mishkan, shortly after the Mishkan was dedicated. The story was interrupted since then with the previous two parshas discussing various kinds of spiritual impurity, mainly the spiritual disease of tzoraas.
This parsha is often part of a double parsha, made up of parshas Acharei Mos and Kedoshim. However, this year we are reading this parsha by itself.
In the first aliyah the story continues from where Aharon's sons just passed away by doing a service in the Mishkan without authorization. G-d here warns Aharon not to approach the ark at the wrong times, and tells him specifically to only approach it on Yom Kippur. This aliyah gives further details of the Yom Kippur service regarding Aharon's changes of clothes and all the times he toivels (immerses) in the mikvah between each change of clothes, and the animals used in the sin offerings.
The second aliyah continues the description of Aharon's duties for the service in the Mishkan on Yom Kippur. Specifically included here is the scape goat ceremony.
In the third aliyah, G-d states that Yom Kippur shall be on the tenth day of the month of Tishrei. This is also the day that Moshe rabenu came down from mount Sinai with the second set of luchos (tablets) and informed the people that they had received atonement for the making of the golden calf. It is a Chassidic principal that we don't celebrate Jewish holidays only to commemorate historical events, but the spiritual revelation that happened to cause the original event happens again, every year on the same Hebrew date. Thus, the prototype Day of Atonement was when G-d forgave the Jewish nation for the making of the golden calf, and the same potential for atonement is repeated every tenth of Tishrei thereafter.
This aliyah states that on Yom Kippur work shouldn't be performed, and that we should afflict our souls. The oral law defines afflicting one's soul as abstaining from eating, washing, anointing, wearing leather shoes, and marital relations.
In the fourth aliyah G-d says that sacrifices may not be offered anywhere else, other than the Mishkan.
In the fifth aliyah G-d says that it is forbidden to eat blood, or animals that are torn in the field or die by themselves.
In addition, when the Jewish people get to the land of the Canaanites, they shouldn't follow after their ways.
Beginning with this, the sixth aliyah, the Torah lists several prohibited sexual relationships with close relatives, and also with a woman who is niddah (who has not yet gone to the mikvah as described in the laws of Taharas Hamishpocha). This section of the Torah is read as the Torah reading at Mincha on Yom Kippur.
The seventh aliyah discusses the prohibition of homosexuality and of bestiality of men and woman. G-d says the Jewish people shouldn't defile themselves with these things as the nations being cast out before them in the land of Canaan defiled the land with these actions. The Torah adds that that is why the land vomited them out. Rashi says that just as a person can't keep vile food in his stomach, the land of Israel can't contain transgressors.