Short answers, pulled from the story.
Taṇhā is a Pali term translated as craving, desire, longing, or greed, whether physical or mental. Scholar Walpola Rahula described it as "the most palpable and immediate cause" of dukkha, the suffering Buddhism identifies as pervading conditioned existence. It appears in the Buddha's second Noble Truth as the origin of suffering.
The Buddha identified three types of taṇhā. Kāma-taṇhā is craving for sensory pleasures; bhava-taṇhā is craving for existence or being, linked to the wrong view of eternalism; vibhava-taṇhā is craving for non-existence, linked to the wrong view of annihilationism.
Taṇhā derives from the Vedic Sanskrit tṛ́ṣṇā, which traces to the Proto-Indo-Iranian tŕ̥šnas and the root tarś- (thirst, desire, wish), ultimately descending from the Proto-Indo-European ters-, meaning "dry." Cognate forms include Old High German durst, Gothic þaursus, Lithuanian troškimas, and Ancient Greek térsomai.
Taṇhā appears in the second Noble Truth, where the Buddha identifies it as the origin of suffering. The third Noble Truth states that the cessation of taṇhā is possible, and the Noble Eightfold Path is the prescribed route to that cessation.
Taṇhā and chanda are both Pali words for desire but are treated differently in Buddhist teaching. Bahm defines taṇhā as "desiring more than will be attained" and chanda as "desiring what, and no more than, will be attained." Peter Harvey notes that chanda can be either wholesome or unwholesome, whereas taṇhā is consistently treated as a cause of suffering.
Taṇhā ends through following the Noble Eightfold Path, particularly through the practice Kevin Trainor calls "insight meditation." This practice cultivates "right mindfulness," which includes understanding three marks of existence: dukkha, anicca (impermanence), and anatta (non-self). Once the non-self doctrine is genuinely accepted, there is, as Trainor puts it, "no locus for clinging," and taṇhā ceases.