The result of an emotion so great that causes one to cry. Usually related to Avatar: The Last Airbender series.
In the series, certain people have special abilities to manipulate elements, which is called bending. These are specific to individual tribes and are waterbending, earthbending, firebending, and airbending. While it takes years of discipline to learn how to bend, even babies can tearbend. Tearbending is a joke the writers created meaning "to cry."
Tearbending made it's debut in the episode, The Ember Island Players.
Actress Katara: *Gasps* An Airbender! My heart is so full of hope, that it's making me tearbend! *Cries*
Actor Sokka: My stomach is so empty, that it's making me tearbend! *Cries*
If they delay the release of Legend of Korra again, I'm gonna start tearbending!