The term "electrocution" is, as
jt says, a combination of the words
electricity and
execute. The term was coined in 1887, and was the result of a competition held by the state of
New York to find a name for their new method of execution. Runners-up were "electromort" and "electricide".
Thomas Edison, who had been hired by the state of New York to develop an electrical method of execution, wanted the process to be known as "Westinghousing".
Edison had used George Westinghouse's alternating current system for his electrocution device as a PR scheme to demonstrate that Edison's direct current system was the safer of the two. Edison's patented DC system, which had the monopoly on electrification in New York City, was being challenged by Westinghouse's more efficient (in terms of delivery) AC system. Edison was quite aware of the superiority of AC and tried to play the "public safety" card, citing the danger of high voltage lines, in an effort to block its acceptance.
The first execution using the method of electrocution took place in August of 1890. It was, reportedly, horribly botched. Westinghouse himself is quoted as saying "they could have done it better with an axe."
Information stolen from Gotham: A History of New York City to 1898 by Edwin G. Burrows and Mike Wallace (a different Mike Wallace).