For most of us, our email inbox is a noisy, hectic place. And plowing through emails can be like hacking through an overgrown jungle of meaningless messages from social networks, shopping websites, listservs, strangers and spammers. Locating a message that matters from your boss, friends or family can feel a bit like unearthing buried treasure. Dozens of apps have tried to solve this problem, and dozens of apps have fallen short. But Google’s Gmail team says it has built a better way, a new app called Inbox by Gmail, which is launching today on Android, iOS and the Web.

“We...