In St. Louis, Missouri, Nikola Tesla made the first public demonstration of such a system in 1893. Addressing the Franklin Institute in Philadelphia and the National Electric Light Association, he described and demonstrated in detail the principles of wireless telegraphy. The apparatus that he used contained all the elements that were incorporated into radio systems before the development of the vacuum tube.
The later derived system to achieve widespread use was 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. For instance, in 1832, James Bowman Lindsay gave a classroom demonstration of wireless telegraphy to his students. By 1854 he was able to demonstrate transmission across the Firth of Tay from Dundee to Woodhaven (now part of Newport-on-Tay), a distance of two miles. 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 Morse code in the tones made by rotary spark gaps.
Canadian-American scientist Reginald Aubrey Fessenden was the first to wirelessly transmit a human voice (his own). Read more about History of radio.
The ultimate development of wireless telegraphy was telex on radio. The most advanced form (CCITT R.44) automated both routing and encoding of messages over short wave radio. Telex on radio was invented in the 1940s, and was for many years the only reliable way to reach many distant contries (See telegraphy for more information).
A good source of history is the book Syntony and Spark: the Origins of Radio, Hugh G. J. Aitken, ISBN 0471018163.