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?
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)
At 2nd level, a titan mauler may choose to wield a two-handed melee weapon in one hand with a –2 penalty on attack rolls while doing so. The weapon must be appropriately sized for her, and it is treated as one-handed when determining the effect of Power Attack, Strength bonus to damage, and the like.