Is it possible to wield a two handed weapon in one hand? If so, what penalties apply? If not, are there circumstances or abilities that would make it possible?
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It is not generally possible
Absent any special feat, class ability, or what have you, it is not generally possible to wield a two-handed weapon in one hand. You can carry one with a single hand, but not fight with it.
Larger creatures can weild smaller creatures’ two-handed weapons in one hand
You can wield the two-handed weapon of a creature one size category smaller than yourself at a −2 penalty. That means you, if we assume you are a Medium creature, could wield a Small two-handed weapon one-handed at a −2 penalty. On average, you generally see a ~1 damage increase for a two-handed weapon over a one-handed weapon, which is not worth a −2 penalty.
Unfortunately, most feats aimed at allowing you to use inappropriately sized weapons without penalty focus on wielding larger weapons. I cannot find any in Pathfinder that work. In 3.5, there were the “web enhancement” kobolds, but as far as I know the Slight Build feature, unique to those kobolds, was literally the only thing that could accomplish it.
Wielding weapons with both hands is typically a good thing
Really must be mentioned: because of the 1½ Str to damage, wielding weapons with both hands is simply better than wielding it in one hand, typically even accounting for the loss of shield or a second weapon.
There may be reasons other than damage that you’re interested in this, but I wanted to point it out.
Add on to KRyan's wonderful answer.
The only class that can wield 2 handed weapons in 1 hand is the Titan Mauler (Barbarian archetype)