Urban myth and incorrect interpretations of the double jeopardy clause would have us believe that the condemned is allowed to go free if an execution fails. However, the urban myths are just myths, and double jeopardy only applies to prosecution, not the carrying out of a sentence once someone has been found guilty.
There's no free ride if the electric chair, the gallows, or a lethal injection doesn't get the job done the first time around. The authorities just have a second go at it. The language of a death sentence is always very clear on this matter and will stipulate that the condemned suffer the given method of execution "until dead."
Even though we've moved beyond firing squads and hangings, botched executions and second attempts still occur. In 1984, the state of Georgia executed Alpha Otis O'Daniel Stephens for the murder of a man who interrupted Stephens during a burglary 10 years earlier. The first jolt of electricity from the electric chair failed to kill him, and Stephens sat in the chair, alive and, by some accounts, semi-conscious, for six minutes while his body cooled enough for doctors to examine him and decide another jolt was needed.
As you can imagine, situations like this can get a little grisly. The Floss isn't in the business of gore, but if you're curious, the Death Penalty Information Center has a list of executions gone awry.