A climate threshold is a critical limit where a climate system responds drastically when exposed to an external forcing, resulting in the system changing into a different stable state. An analogy can be made to tipping a wine glass - if it tips too far from its standing state it will fall and spill the wine, leaving the glass in a stable state on its side. Thus a climate threshold is also referred to as a "tipping point". Climate threholds are an example of how the climate behaves in a non-linear way, often as a result of positive climate feedback mechanisms.
The climate tends to respond to changes in a gradual way until it reaches a threshold. Once past this threshold, what is known as abrupt climate change begins to occur, where the climate's response to a forcing is much larger than the change in that forcing. Thresholds are important to study in order to test how well climate models can simulate their effects.
For example, a threshold exists with the increased amounts of freshwater flowing into the ocean from melting ice. This freshwater is a potential forcing for abrupt climate changes; both sea levels and ocean circulations could change drastically.