Senators of the United States Congress were originally chosen by state legislatures. Citizens would vote for their state legislators, and those legislators would vote a man into the U.S. Senate.
At the beginning of the 20th century, though, many states had begun to use the popular vote to elect U.S. Senators. But it wasn't a direct election; the election appeared on ballots as a referendum, and the results of that referendum were then confirmed by the state legislature.
In 1913, however, the Seventeenth Amendment to the U.S. Constitution standardized the popular election of Senators throughout the entire country. The Seventeenth Amendment also gave a state's acting governor the power to appoint someone to the U.S. Senate in the event of a vacancy.
Because this amendment gives a single individual the power to appoint a U.S. Senator, many states' rights advocates have called for a repeal of the Seventeenth Amendment. They argue that this gubernatorial power bypasses the democratic process.
Although it's easy to see how such power could be misused (consider the recent scandal involving Illinois governor Rod Blagojevich), it can be wielded to create change for the better. For example, five of the first six women to serve in the U.S. Senate were appointed to that position by their state governors.