The Good Samaritan Rule is named after a story in the Bible in which a Samaritan (an ethnicity of people the Jews generally disliked at the time) helps a Jew in a time of need, thus making him the "good Samaritan". It says essentially that we have some obligation, either morally or legally, to help others who need help. When you ask people, most of them agree with something similar to this; if you can help without great cost to yourself, you ought to do so.
The American Bystander Rule, on the other hand, is a legal doctrine in common law (originating in the US, hence the name) that bystanders do not have any legal or criminal liability for failing to help someone in need, even if they could have done so very easily. One of the arguments in its favor is that people already tend to help out of a feeling of moral obligation; but if they don't (and some people don't!), the American Bystander Rule requires that they cannot be punished for their inaction.
Both principles prescribe approaches to dealing with incidents where someone helps or doesn't help someone else; but they prescribe fundamentally opposite actions. The Good Samaritan Rule says that we must help; the American Bystander Rule says that we do not need to help, and no one can punish us if we don't. It is possible to believe in the Good Samaritan Rule morally but the American Bystander Rule legally, on the grounds of personal liberty: Perhaps the government has no right to compel you to act, even if morally you should.