This parsha continues to tell us what occurred after Parshat Shmini (of two weeks ago) where Aharon's two sons died when they offered an unauthorized offering in the Mishkan.
The story was interrupted since then with the previous two parshas discussing various kinds of impurity, mainly the spiritual disease of tzoraas.
This is a double parsha, made up of parshas Acharei Mos and Kedoshim. The seven aliyot of this week's reading are indicated by special aliyah markings within those two parshas.
In the first aliyah the story continues from where Aharon's sons just died 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 immerses in the mikvah between each change of clothes, and the animals used in the sin offerings.
In the second 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 Rabeinu 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's a Chassidic principal that we don't celebrate Jewish holidays only to commemorate historical events, but the revelation that happened for 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 third aliyah G-d says that sacrifices may be offered nowhere else, other than the Mishkan.
G-d also says that it is forbidden to eat blood, or animals that are torn (in Hebrew: "treifa") or die by themselves.
In addition, when the Jewish people get to the land of the Canaanites, they shouldn't follow after their ways.
The Torah then goes on to list several prohibited sexual relationships with close relatives, and also with a woman who is nida (who has not yet gone to the mikvah, with all its particulars).
The fourth aliyah contains the end of parsha Acharei and the beginning of parsha Kedoshim and discusses the prohibition of homosexuality and of bestiality of men and woman. G-d says that 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's 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.
Parsha Kedoshim then begins. G-d commands Moshe to admonish the Jewish people to be holy, then summarizes several laws, such as: fearing parents, observing Shabbos, leaving over parts of the harvested field for the poor, not to lie, not to put a stumbling block before the blind, and others.
The fifth aliyah contains an assortment of commandments: to judge in righteousness, not favoring either the poor or the rich, not to go around as a talebearer, not to stand idly by when your neighbor is in danger, not to hate your fellow in your heart, to surely rebuke your neighbor, not to shame him in public, not to take vengeance, and not to bear a grudge. And in this aliyah is the commandment to love your neighbor as yourself (which Rabbi Akiva said is a fundamental principle in the Torah). Chapter 32 of the Tanya is an essential lesson on these matters.
The following statutes are also given here: not to sow a field with two kinds of seed, and not to wear a garment made of a mixture of wool and linen (which is called shatnez). The checking of a garment for this forbidden mixture is called shatnez testing.
Also here: the law of "orla," forbidding the fruit of the first three years of a tree, and not to mar the corners of the beard - i.e. not to use a razor blade to shave off facial hairs.
The sixth aliyah gives the commandment to not harm a converted person; in fact we are commanded to love such a person as we love ourselves.
Furthermore, here we are commanded here to be honest in various weights and measures.
The seventh aliyah begins with the prohibition against cursing parents, then the prohibitions of adultery and other forbidden sexual relationships including homosexuality, bestiality, and incest, and relations with a woman who is nida (hasn't properly gone to the mikvah).
Finally, the aliyah ends with G-d telling us that he has given us a land flowing with milk and honey, and he has separated us from the nations with his commandments. By doing the commandments we connect to G-d and are sanctified and holy which is what the name of this parsha, "Kedoshim", means.