Short answers, pulled from the story.
Vedanā is a Pāli and Sanskrit term traditionally translated as "feeling" or "sensation." It refers specifically to the pleasant, unpleasant, or neutral affective tone that arises when a sense organ comes into contact with a sense object and the associated consciousness. It is not the same as emotion, which Buddhist teachers describe as a more complex phenomenon involving many mental factors.
Bhikkhu Bodhi distinguishes vedanā as "the bare affective quality of an experience" rather than a full emotion. Chögyam Trungpa Rinpoche writes that what we ordinarily mean by hurt feelings belongs to the fourth and fifth skandhas, not to vedanā. Vedanā is only the raw valence of experience: pleasant, unpleasant, or neutral.
In the chain of dependent origination, vedanā arises from contact (phassa) and in turn conditions craving. The fifth-century Visuddhimagga clarifies that while vedanā is decisive in leading toward craving and suffering, the conditional relationship between feeling and craving is not karmically necessary, meaning the practitioner can interrupt it.
The Pali canon describes as few as two kinds of vedanā (physical and mental) and as many as 108. The most common scheme is threefold: pleasant, unpleasant, and neutral. The Visuddhimagga favors five types, while other canonical discourses enumerate six, eighteen, thirty-six, and 108 kinds by multiplying the basic categories across sense faculties and time periods.
Vedanā is one of the four foundations of mindfulness (satipaṭṭhāna), the objects of focused attention in Buddhist meditation. The canon states that meditating with concentration on vedanā can develop deep mindfulness and clear comprehension of impermanence and attachment. The Satipaṭṭhāna Sutta and the Ānāpānasati Sutta both address this practice directly.
Vedanā is recognized in both traditions. In Theravāda Abhidharma it is one of seven universal mental factors; in Mahāyāna Abhidharma it is one of five. In both traditions it also appears as one of the twelve links of dependent origination and as one of the five skandhas.