(With examples from Discworld.)
This topic was hinted at in an entry on irregular verbs, and I'm sure you've all been waiting anxiously for follow-up ever since.
The answer is (drum roll, please): both.
"Hang" is one of the few verbs with both a regular and irregular past tense form that has rules governing when each should be used. The most simple explanation given is that people are hanged, while pictures (and anything else) are hung. And really, that will get you a fair way if it's all you remember. But it is a little more nuanced than that: It's true that only people are hanged, but people can also be hung.
According to my usage guru Bryan Garner, "hanged is to be suspended by the neck with a rope or cord for the purpose of causing death." (Garner's Modern American Usage)
From the darkened window of her velvet-lined carriage, Countess Schuei eyed the hanged man with distaste. You would never see such a thing at her Überwald estate. Even as nobility she had been taught not to waste food.
But what if you're talking about a person (not a picture) being suspended, but the rest of the criteria doesn't apply?
Vimes nodded, amiably enough given the circumstances, to the assassin who was hung upside down by one foot in a snare near the outer courtyard. He supposed he would cut the fellow down after lunch; a few hours of blood settling in his head might help him realize the folly of making another attempt.
Even though it's a person being suspended, it is not by the neck for the purpose of death, so hung is correct here.
We would also find hung in figurative uses involving people, such as a person being hung out to dry.
Of course, as with many rules this limited and specific, there are those who maintain that the hanged distinction is archaic and should be done away with; that hung is perfectly acceptable for all circumstances and will be what we use in another fifty years anyway. They may be right in the long run, but we're not there yet, and personally I'm glad of it. I find the specificity of hanged to be in keeping with the severity of its definition. I feel a chill in the pit of my stomach when I hear it, and I think it would be a shame to lose that.
Besides, on a less serious note, I remain in favor of any and all opportunities for juvenile humor.
Magrat set aside Nanny Ogg's tarot deck in embarrassed consternation. She was certain it was supposed to be "The Hanged Man."