The article is a little confusing, in that it says the chance of a solar flare as bad as the "Carrington Event" of 1859 has a 12% chance of hitting in any given decade, and doesn't explain the obvious arithmetic paradox. Meanwhile, based on slightly older research, that Wikipedia article I linked suggests a frequency of once every 500 years.
The trigger for the discussion is a solar flare that was that bad in 2012, only not pointed straight at our Earth. http://www.forbes.com/sites/ericmack...-missed-earth/ looks like a guide to some earlier discussion of the same flare.
The effect would apparently be to take out most of the satellite grids (comm sats, GPS), set some of the terrestrial electronic grid on fire, cause some hellacious power spikes for computers (perhaps starting some additional fires), and so on. http://en.wikipedia.org/wiki/Geomagnetic_solar_storm has more, suggesting that electricity might be out for weeks.
Obviously, greater solar flares could be even worse, up to and including depopulating the planet. Presumably the utterly catastrophic ones are quite unlikely; certainly I don't recall any claims of scientific evidence of them occurring with much frequency over the past few billion years.
This is also the premise of one of the great science fiction stories, Niven's Inconstant Moon. http://downtownmagnets.edlioschool.c...ant%20Moon.pdf It's from the 1960s, which is why there are no internet nor cell phones in it.