Two-handed weapons are wielded with both hands. Generally speaking, any weapon which is one size category greater than the character must be wielded two-handed. A character will also wield any weapon of the same size category as the character two-handed if the off-hand is empty. Smaller weapons are never used two-handed.
Most of the player races are medium sized, and so any large weapon such as greatswords or spears must be wielded with both hands, and any medium weapon like a morningstar can be wielded two-handed unless the off-hand is also used. Gnomes and halflings are small sized, so large weapons are off-limits and all medium weapons must be held two-handed.
Wielding a weapon with both hands results in a 1.5x bonus on Strength (or Intelligence, with Combat Insight) modifiers to damage rolls. Thus, two-handed weapons are more damaging than a weapon used only in the main hand.