I must say, before I begin, that Abraham was under the laws of Noah, with the addition of circumcision (non-binding on people who are not part of the covenent of Abraham). Okay, here goes.
First of all, the number of laws = the number of colors easily visible in a rainbow, and also the number of days of creation, that is, seven.
This is show from a number of incidents in Genesis, including the fact that G-d created the world. We see the negative consequences of denying G-d in the Tower of Babel incedent (Gen 11-they wanted to make a name for themselves, as if they were as powerful as G-d).
Since G-d created the world, this is a given, but it is explicity stated in Lev 24:16 "And he who utters the name of the L-RD blashphemously...both the stranger and the resident alien..." The stranger is a non-Jew (though he/she may be in the process of learning about Judaism and converting, but not yet Jewish).
Cain and Abel (Gen 4:8-16) is an obvious example of murder, and G-d clearly does not approve of it, but there is a specific prohibition as well. Gen 9:5-6 "But for your own life-blood [human life] I will require a reckoning: I will require it of every beast; of man too, I will require a reckoning for human life, of every man for that of his fellow man! Whoever sheds the blood of man, by man shall his blood be shed; for in His image did G-d make man." The fact that man will shed the blood of the murderer, also leads to the necessity of courts (law 7).
Also included under this law are such things as not fighting wars of agression (self-defense is permitted), not to abort a fetus if the mother's life or reproductive ability is not in danger, no euthenasia or assisted suicide,and no suicide. A child of Noach may not break the law not to murder even to save their own life. For example, to murders another because you yourself are being threatened by another at gunpoint, is prohibitted--a person may defend themself, but may not murder a person who is not out to murder them. The other 6 laws may be violate in order to save your life, but in no case may you murder.
Ham's son Canaan had incestuous, homosexual relations with his grandfather, Noah. (Gen 10:20-28) In the Tanach, to uncover someones nakedness it to have sex with that person. So, Canaan must have had sex with Noah while Noah was drunk. Since Canaan got punished, not Ham, Canaan must have done the deed. Later the Torah, it will say that if you uncover the nakedness of your mother, for example, it is as if you were uncovering the nakedness of your father. Thus, Ham is connected in the verse, by the deed was done by Canaan. Additionally, there is the incident with Lot and his daughters, which was incestuous.
Adultary is shown to be prohibited by G-d's reaction when Pharaoh takes Sarai into his house(Gen 12:10-20, see also Gen 20:1-7).
Bestiality. I am not really sure where this comes from, perhaps this is infered from the other prohibited sexual acts, perhaps not. Do any of you know where this is (pre-Sinai, or is it another case of the law applying to a stranger)? One reader of this page suggested it may stem from where G-d shows Adam all the animals, but Adam was alone. G-d created then Eve, but it is clear from this that humans and animals are not compatible in this way.
Additionally, it says in Genesis 2:24 "Hence a man leaves his father and his mother and clings to his wife, so that they become one flesh". Clearly he is not to be one flesh with anyone except his wife.
There is the incident where Joseph's brothers find thier money back in their sacks. They are dismayed, clearly they are afraid they will be accused of stealing (Gen 43:35). Again, in Gen 44:1-10, notice that they were accused of stealing in verse 8, that stealing is evil (verse 4), wicked (verse 5), and that they brothers expected that if one had stolen, that they should die (verse 9) and the rest become slaves. The Egyptian says that only the one who has the goblet will become a slave. Clearly stealing is illegal.
Besides the obvious acts of theft, this also means that a B'nai Noach cannot: cheat a worker or employer out of money, overcharge, shift a landmark to add someones property to your own, refuse to pay money owed, kidnapp a person (this, by the way, includes a prohibition to kidnapp a fellow to enslave them), use false weights or mesures, looting in wartime (or in peacetime, as that too is theft), or committ rape.
Gen 9:3-4 "Every creature that lives shall be yours to eat; as with the green grasses, I give you all these. You must not, however, eat flesh with its life-blood in it." Flesh with its life blood in it is considered to be eating from a living animal.
Gen 7:11 "The earth had become corrupt before G-d; the earth was filled with lawlessnes" Additionally, this law allows Noachide communities to make other laws their society needs, provided they do not confilct with the first 6 laws above.
For more information, please visit Chavurath B'nai Noach's Home Page