Dora Bas Rivka Silver O'H
G-d told Noach to take his family and the animals out of the ark. The animals were to be released to breed and replenish their populations, minus those surplus kosher animals that Noach had brought to offer as sacrifices of thanks to G-d. Noach's actions pleased G-d, Who said that He no longer curse the world because of mankind. People have a natural inclination to do evil, so He will never again smite the entire planet in one fell swoop, nor would He suspend the natural order of the universe, as was done during the course of the deluge.
G-d blessed Noach and his family that they should be prolific and refill the Earth. He gave them dominion over the other creatures of the world, with the permission to eat meat. (Adam and his descendants had been vegetarian until that point.) They were prohibited, however, from eating flesh torn from a living animal.
While the descendants of Noach were given permission to eat animals, they were cautioned against murder and suicide. (The Talmud in Sanhedrin 57b applies this prohibition even to abortion.) Murder would be considered a capital crime, as man was created in the image of G-d.
All told, there are seven "universal laws," which were commanded to the descendants of Noach (that is, everyone in the world). In no particular order, these are: (1) Not to kill, (2) not to steal, (3) not to eat a limb from a live animal, (4) not to worship idols, (5) not to curse G-d, (6) sexual morality and (7) to establish courts of justice. Each of these is actually a category comprising several laws. For example, sexual morality includes prohibitions against adultery, incest and more. Not to steal includes prohibitions against burglary, robbery, kidnapping and rape. (Rape falls under theft rather than sexual morality because the relationship is one that might be permitted if consensual; it is the coercion that is prohibited in this case.) The Talmud in Chulin 92a says there are 30 such individual laws.