The invention of the sneaker was pretty much a direct result of the discovery of rubber vulcanization (Charles Goodyear), and the invention of the lock-stitch sewing machine (Elias Howe). One of the applications of vulcanized rubber that had huge popularity after its development was waterproof boots. Although these boots were very good for times when it was raining, or when one had to slog through muddy fields and paths, they were heavy, and quite warm.
Charles Goodyear's son, Charles Jr., was aware of the problems inherent with the design of the boots, and for some time tried to find an inexpensive way to affix light canvas tops to the waterproof rubber shoe bottoms. When he learned of his friend Elias Howe's success with the sewing machine, which was able to sew thick pieces of leather with ease, he knew he had found the solution.
The first pair of sneakers, called plimsolls, were manufactured in the 1870s, and have enjoyed much success, and many revisions since that time.
The term "sneaker", though, didn't come into use until about 1917, when U.S. Rubber began producing a shoe called Keds. The name sneaker came from the shoe's quiet motion, allowing someone wearing Keds to sneak up on other people. The word was coined by a marketing man for N. W. Ayer & Sons, Henry Nelson McKinney.
In modern parlance, sneakers may also sometimes be called tennis shoes. In the UK, they may still be referred to as plimsolls, or trainers.
Buchman & Groves, Fifty discoveries that changed the world. Scholastic, 1988.