In chapter 23, Atticus and Jem are discussing the conviction of Tom Robinson, and Atticus "assured" the kids that Tom wouldn't be executed until a higher court had a chance to review his case. That is what is called an appeal. An appeal only comes after the first trial and conviction are over and judgment has been filed. While Tom waits for the appeal process to begin, he is sent to Enfield Prison Farm in Chester County. Unfortunately, Tom Robinson never gets to appeal his conviction because he is shot attempting to escape from prison during the inmates' exercise time. Tom takes his chances by sprinting for the fences rather than waiting for an appeal to make its way from Atticus's desk to a higher court. Appeals can take years to complete. As a result of Tom's decision to make a run for it, he is shot by the guards, and the chance for an appeal to save him is never even given the chance to begin.
Tom was sent to a prison camp while Atticus was appealing his case, but Tom gave up hope, tried to escape, and was shot and killed.
After the trial, Tom was taken to Enfield Prison Farm, which was about 70 miles away. This is where he would attempt to escape and be shot to death.
His chances were slim to none that he would have won his appeal, and he would have been sentenced to die at that point. Tom took his chances at the only chance of freedom he would have, although surely he must have realized he would not have much chance of being successful at an escape.