The sins that cry to Heaven (or sins that cry Out to Heaven) is a group of sins in Catholic doctrine that cuts across mortal sins and venial sins. All of them are exemplified by a biblical sin:
* Willful murder – Cain's murder of his brother – Genesis 4:1-16
* Sodomy – the sin of Sodom – Genesis 19:5
* Oppression of the poor esp. widows, orphans and strangers for which the Catechism of the Catholic Church cites Exodus 20:20–22 (which is not obvious to the non-believing).
* Defrauding laborers of their wages – based on Deut 24:14–15
The list is just a list of sins which in the Bible are mentioned as crying out to the Lord for justice. The Catechism does indeed cite Exodus 20:20-22 as the source for oppression of the poor; and no doubt that passage doesn't obviously support the point for the non-believing, because it doesn't support it at all, whether you are non-believing or not. It's a typo; the passage in view is Exodus 22:20-24, which is a quite grim picture of divine reciprocity:
You shall not wrong or oppress a resident alien, for you were aliens in the land of Egypt. You shall not abuse any widow or orphan. If you do abuse them, when they cry out to me, I will surely heed their cry; my wrath will burn, and I will kill you with the sword, and your wives shall become widows and your children orphans.
To say that the sin of Sodom that cried out to the heavens for vengeance was the sin of sodomy is more than a bit of understatement; the context in Genesis is very clear that the men of Sodom were gang-raping travelers. The sequence of events starts with the Angel of the Lord telling Abraham that people are crying out to heaven against the sins of Sodom, and He says that he is going down into the city to see if it is true: "I must go down and see whether they have done altogether according to the outcry that has come to me; and if not, I will know." And then the two angels go down to Sodom to stay at Lot's house, where the men of Sodom gather, demanding that Lot give them the two guests to rape -- thus showing not only that they are sexually dissolute but that their sexual depravity extends even to disregard of the (at the time) iron-clad moral conventions concerning the basic hospitable treatment of guests. And then the angels tell Lot to flee the city, because it will be destroyed; the actions of the men of Sodom have confirmed the outcry to heaven against them. Lacking a real sense of hospitality, we lack a proper sense of the horror the ancient world would have had of the sheer, terrible sexual depravity involved in such violation of hospitality. But, no doubt, we can still see how it might cry out for vengeance.
The final sin in the list, robbing workers of their wages, is interesting in that it is mentioned in both the Old Testament and the New; and, indeed, the picture in James 5 is even more vivid than that found in Deuteronomy:
Come now, you rich people, weep and wail for the miseries that are coming to you. Your riches have rotted, and your clothes are moth-eaten. Your gold and silver have rusted, and their rust will be evidence against you, and it will eat your flesh like fire. You have laid up treasure for the last days. Listen! The wages of the labourers who mowed your fields, which you kept back by fraud, cry out, and the cries of the harvesters have reached the ears of the Lord of hosts. You have lived on the earth in luxury and in pleasure; you have fattened your hearts on a day of slaughter.