According to menshealth.com
The very first sneaker was invented by Wait Webster in the 19th century. Of course, it wasn't called a sneaker then—and it looked very different. Plimsolls, as they were called, featured thin rubber soles, an equally thin material on top, and were the preferred footwear for active activities.
Also wikipedia have some interesting info on this subject as well.
These shoes acquired the nickname 'plimsoll' in the 1870s, derived according to Nicholette Jones' book The Plimsoll Sensation, from the coloured horizontal band joining the upper to the sole, which resembled the Plimsoll line on a ship's hull. Alternatively, just like the Plimsoll line on a ship, if water got above the line of the rubber sole, the wearer would get wet.
Plimsolls were widely worn by vacationers and also began to be worn by sportsmen on the tennis and croquet courts for their comfort. Special soles with engraved patterns to increase the surface grip of the shoe were developed, and these were ordered in bulk for the use of the British Army. Athletic shoes were increasingly used for leisure and outdoor activities at the turn of the 20th century - plimsolls were even found with the ill-fated Scott Antarctic expedition of 1911. Plimsolls were commonly worn by pupils in schools' physical education lessons in the UK from the 1950s until the early 1970s.
British company J.W. Foster and Sons designed and produced the first shoes designed for running in 1895; the shoes were spiked to allow for greater traction and speed. The company sold its high-quality handmade running shoes to athletes around the world, eventually receiving a contract for the manufacture of running shoes for the British team in the 1924 Summer Olympics. Harold Abrahams and Eric Liddell won the 100 m and 400 m events, kitted out with Foster's running gear.
For more information look at this article: https://en.wikipedia.org/wiki/Sneakers#History