I'd have to say the invention of the transistor was the most transformative thing to happen to society. Prior we had vacuum tubes sure but they were power hungry devices that made portable electronics impossible.
The transistor changed everything. It also allowed the device on which I'm posting to come about. A device admittedly a bit dated already but still enough to allow me to multi-task, listening to music, watching video, etc. And to continue the line of thoughts - the computer has invaded every aspect of life itself. All because of the invention of the transistor.