NEW DELHI—The hottest phones for the world’s next billion users aren’t made by smartphone leaders Samsung Electronics Co. or Apple Inc. In fact, they aren’t even smartphones.

Millions of first-time internet consumers from the Ivory Coast to India and Indonesia are connecting to the web on a new breed of device that only costs about $25. The gadgets look like the inexpensive Nokia Corp. phones that were big about two decades ago. But these hybrid phones, fueled by inexpensive mobile data, provide some basic apps and internet...