Frank Darabont, who also directed the movie versions of King's The Shawshank Redemption (1994) and The Green Mile (1999). The Mist is way different from Shawshank, mind you...
This is a technically well-plotted horror film. The mist of the title sweeps across a small town in the mid-West (King loves his American small town settings), and catches a father and son in a supermarket picking up supplies. Most of the action takes place in and around the supermarket over a period of less than 24 hours, and deals with the disparate, random group of civilians (and three soldiers) caught by the mist, and the creatures that live within it.
There is a whiff of the post-apocalyptic in this story: the townsfolk are very much kept in the dark about the mist, its origins, and what else could be lurking in it. But they also begin to speculate about the extent of the mist, and why rescue has not come. It is this fear that the mist spells the end of the world and of humanity that influences the actions of some of the characters - for instance the beginning of a cult within the group of survivors, as they splinter into factions and argue over the best means of survival (many conflicts degenerate into whether to escape from the supermarket or stay and try to defend it).
The Mist is less about the horrors without and more about the horrors within: how do ordinary people behave when the veneer of civilization is suddenly stripped away and they are faced with the very real possibility of death? How fast is it for superstition and religion to take hold of the seemingly rational 21st century mind? And rather than look at the behaviour of one group with strong ties - like a platoon of soldiers or a family - the film manages to engineer a situation where a random cross section of normal folk are caught in one place, namely a supermarket, somewhere where people habitually go and where at any moment in time a single snapshot of local humanity can be caught - in this context, literally.