Breakthroughs that would change the way the world uses power. Orders of magnitude advances in battery capacity, charging time, power generation and consumption efficiency. If our houses used 10% of today's power, and that power cost 10% as much to generate, you'd be paying less than a dollar per year for all your electricity. Or we'd use 10 times as many appliances, because we could, and that would still cost less than $10 per year.
An order of magnitude advance in battery power and electronics efficiency would mean that your phone could go for about three months on a single charge while still doing everything you do today, and when you do need to charge it up, it would take about a minute. Imagine only needing to charge your phone four times a year, for one minute each time. I expect a lot of new possibilities for mobile computing might open up. We might have more of them or use them in more situations.
Mokalus of Borg
PS - With that level of power storage and use, maybe we wouldn't even have big central power stations any more.
PPS - It might become much more local.