Athletic or casual rubber-soled shoes are called sneakers. Before you walk your neighbor's Great Dane, you'll have to take off your work shoes and put on your sneakers.
You can also call sneakers tennis shoes, kicks, or running shoes, and if you're in Britain, you can call them trainers or plimsolls. Sneakers are made for exercise and sports, but they're also very popular everyday shoes because they're so comfortable. Sneaker, which is most common in the Northeast US, comes from their noiseless rubber soles, perfect for sneaking. Originally, they were called sneaks.