Wireless telegraphy is the practice of remote writing (see telegraphy) without the wires normally involved in any electrical telegraph. The first practical system to achieve widespread use was first demonstrated by Guglielmo Marconi in 1896, though it had roots in earlier work by many scientists and inventors. As far back as Faraday and Hertz in the early 1800s, it was clear to most scientists that wireless communication was possible, and many people worked on developing many devices and improvements. The first wireless telegraphy devices started appearing in the 1860s, many based on the work of Nikola Tesla. Edison, for example, patented one in 1885 for use by trains. Marconi and Braun shared the 1909 Nobel Prize in physics for "contributions to the development of wireless telegraphy".
A few decades later, the term radio became more popular. Early radio could not transfer sounds, only the clicks of Morse code. Canadian-American scientist Reginald Aubrey Fessenden was the first to wirelessly transmit a human voice (his own). Read more about Radio/History.
A good source of history is the book Syntony and Spark: the Origins of Radio, Hugh G. J. Aitken, ISBN 0471018163.