The FAA conducted 10 tests in which they forced a laptop into what's called "thermal runaway," a not-uncommon state in which the laptop battery's temperature continuously rises. In one test, an 8-ounce bottle of dry shampoo was strapped to a laptop and caught fire almost immediately. The shampoo exploded in under 40 seconds.
That's fast enough to beat fire-suppression systems, which could mean what the FAA's report ominously calls "loss of the aircraft."
There's a chance this ban won't include domestic flights since the danger is increased when flying overseas, but we won't know much until the UN considers the proposal in the coming weeks. If the UN decides to move forward with the ban, the International Civil Aviation Organization will set safety standards, but individual countries still have to ratify the rules for themselves. Since the US is the country proposing the ban, it's likely that we'd follow suit.