How does society generally reward and punish in America? Which theory is generally followed, and how well does it work?

Systems of punishment and reward in the US have traditionally relied most heavily upon the theory of retribution, rather than the utilitarian approach which focuses more upon deterrence than upon punishing an offender because it is what he or she deserves.
Utilitarian approaches to crime and punishment seek to render back to society a greater level of good than the level of "bad" inflicted by a crime in the first instance. This usually involves attempts to both deter society as a whole from committing crimes and also to rehabilitate a specific individual so that he himself will not commit further crimes.
Where retribution is the defining element of crime and punishment, as it is in the US, an offender is punished not in order to improve society, or that individual, but because the crime must be equalized, as it were, with a just punishment. This is the "eye for an eye" approach which offers the death penalty as a punishment for murder because the two acts cancel each other out in terms of social balance, as it were. There is, of course, also an element of deterrence in retributive punishment: it does not deter the individual from committing further crimes if he is put to death, but arguably it deters others from committing murder (although really there is very little evidence of this). The US also does incorporate elements of utilitarian theory in terms of rehabilitative programs, which are run for those who are mentally ill, for example, to help them readjust to society upon leaving prison.
There are certainly social benefits to retributive punishments. Polls have shown that the public likes to see that criminals are punished: this satisfies their desire to see justice done. People very often vote for candidates for political office who promise to be "tough on crime." As such, retributive punishment does help to keep social order, particularly in the US where this is the approach to punishment which has long been preferred.
However, it is notable that the US is the only remaining Western country which uses the death penalty. The reason most countries no longer do this is because it has been repeatedly proven that it does not deter crime. In the US, it is not really being used as a deterrent—the philosophy is that the death penalty is enacted upon an individual because it is what he deserves. But crime rates are not lower in places that have the death penalty than in those which do not, because, as has been repeatedly shown, people do not consider the consequences of murder in the moment of committing it.
Another area in which the US's retribution-based punishment system falls down is on the issue of recidivism. Because the tendency of the system is to consider retribution, rather than utilitarian benefits to society, there is a lack of funding for programs which would minimize recidivism rates, particularly in the mentally ill. Increased funding and efficiency, both while in prison and upon release, to treat prisoners' mental illnesses and help them re-adjust to society would result in lower recidivism rates in the area of drug offenses and domestic violence. However, because the slant of the US justice system is toward retribution, there has been less focus upon this area.
https://law.jrank.org/pages/9576/Punishment-THEORIES-PUNISHMENT.html