The word “pneumatics” typically refers to the use of compressed air to increase the effective power of certain tools. In this sense, ancient blacksmiths designing their own bellows may have invented pneumatics.
Since the word “pneumatic” refers to anything containing wind, gases, or air (particularly compressed air), we could say that the first humans to use a blow dart gun invented pneumatics. According to the more common sense of the word, used to refer to pneumatic tools, we could rather say that the early smelters and blacksmiths that were the first to use hand bellows, which could be regarded as pneumatic tools, invented pneumatics.
Tinkerers and engineers modified and improved the basic hand bellows design until they’d invented mechanical air compressors. Although German physicist and engineer Otto von Guericke's attempts to produce an atmospheric vacuum improved on17th-century air compressors, it wasn’t until 1829 that the first compound air compressor was patented. By 1872, the use of water jets had improved compressor efficiency and paved the way for the invention of water-jacketed cylinders.
Although some mistakenly afford John Dunlop the distinction of inventing the pneumatic tire in 1892, (See Reference 3) it was actually the Scottish inventor Robert William Thomson who was the first to patent a pneumatic tire in 1845, dubbing it the “aerial wheel.” Thomson’s “tyre” was a hollow belt of India-rubber inflated with air.
In 1810, the English inventor George Medhurst published an idea to use pneumatic tubes to carry letters and packages. By 1844, the first pneumatic rail service (or the “atmospheric railway”) had been built in Ireland (http://www.columbia.edu/~brennan/beach/chapter2.html), leading to experiments in their use for human passengers; these experiments were abandoned when the steam locomotive was developed four years later. In 1865 in the U.S., Alfred Beach patented his pneumatic tube design, which he then used to build the first pneumatic tube rail service in New York.
Samuel Ingersoll and Charles Brady King are largely credited with inventing the pneumatic drill in 1871 and the pneumatic hammer in 1890, respectively (http://inventors.about.com/od/pstartinventions/a/pneumatic.htm ); however, there seems to be little in the way of credible documentation for these claims.