In nearly all criminal trials in the US, the judge does decide the sentence, although for some charges (drug charges, I think) they are required to follow mandatory federal guidelines.

It was a ruling made by the US Supreme court that is responsible for the two-phase jury trial in capital murder cases. The first phase is the guilt or innocence phase. If a guilty verdict is rendered, then the case proceeds to the penalty phase and the jury will hear evidence of both aggravating and mitigating circumstances. I'm not entirely certain about this, but in at least some states a judge may have the authority to override a jury's imposition of the death penalty under certain limited circumstances. The judge may not override a jury's decision to impose a sentence of life instead of the death penalty.   

Before bringing a murder case to trial the prosecutor must decide and announce if they intend to seek the death penalty. The state where I live specifies the conditions that may be considered aggravating circumstances that warrant a prosecutor seeking the death penalty.

Any potential juror who says they could not consider imposing the death penalty is excused from hearing a capital case.