The Exigents, also called the Exigent Exalted (although this term is technically incorrect - the Exigence itself is a method of Exaltation), are a type of Exalt introduced in Exalted Third Edition and scheduled to be described in full in Exigents: Out of the Ashes.
When a lesser god finds themselves in direst need, they may pray to the Unconquered Sun to be allowed to create an Exalt of their own. If the need is great enough, if there is no other way to deal with the situation, and if the Unconquered Sun deems it worthy, then the god will be granted a portion of the Exigence, the divine flame, so they may choose a champion.
The Exigence does not come without cost. Creating an Exigent requires a god to permanently sacrifice a good part of their power, and in the case of weaker gods it may consume them outright. Nevertheless, some gods deem it worth the price.
The only thing connecting the Exigents is their common source in the Exigence. There is no singular 'type' of Exigent, as there is with other Chosen; instead there are as many types as there are gods who have called upon the Exigence, with their nature and powers reflecting that of the god who chose them. Their strength, too, can vary, from being able to match the Dragon-Blooded to being on par with the Celestial Exalted.
The Exaltations of the majority of Exigents exist only for the Exalt's lifetime. For Exaltations that are inheritable, they may be passed on like those of the Celestial Exalted, like the Dragon-Blooded, or more likely in their own unique manner.
It's possible for the Exigence granted a god to be stolen or traded, perhaps falling into the hands of criminal or forbidden gods, or perhaps being modified by a number of gods to create an aberrant Exalt of incredible power.