Theravada

Bhikkhu Bodhi states:

Feeling is the mental factor which feels the object. It is the affective mode in which the object is experienced. The Pali word vedanā does not signify emotion (which appears to be a complex phenomenon involving a variety of concomitant mental factors), but the bare affective quality of an experience, which may be either pleasant, painful or neutral....[3]

Nina van Gorkom states:

When we study the Abhidhamma we learn that 'vedanā' is not the same as what we mean by feeling in conventional language. Feeling is nāma, it experiences something. Feeling never arises alone; it accompanies citta and other cetasikas and it is conditioned by them. Thus, feeling is a conditioned nāma. Citta does not feel, it cognizes the object and vedanā feels...All feelings have the function of experiencing the taste, the flavour of an object (Atthasālinī, I, Part IV, Chapter I, 109). The Atthasālinī uses a simile in order to illustrate that feeling experiences the taste of an object and that citta and the other cetasikas which arise together with feeling experience the taste only partially. A cook who has prepared a meal for the king merely tests the food and then offers it to the king who enjoys the taste of it:...and the king, being lord, expert, and master, eats whatever he likes, even so the mere testing of the food by the cook is like the partial enjoyment of the object by the remaining dhammas (the citta and the other cetasikas), and as the cook tests a portion of the food, so the remaining dhammas enjoy a portion of the object, and as the king, being lord, expert and master, eats the meal according to his pleasure, so feeling, being lord, expert and master, enjoys the taste of the object, and therefore it is said that enjoyment or experience is its function.Thus, all feelings have in common that they experience the 'taste' of an object. Citta and the other accompanying cetasikas also experience the object, but feeling experiences it in its own characteristic way.[4]

Mahayana

The Abhidharma-samuccaya states:

What is the absolutely specific characteristic of vedana? It is to experience. That is to say, in any experience, what we experience is the individual maturation of any positive or negative action as its final result.[5]

Mipham Rinpoche states:[6]

Sensations are defined as impressions.The aggregate of sensations can be divided into three: pleasant, painful, and neutral. Alternatively, there are five: pleasure and mental pleasure, pain and mental pain, and neutral sensation.In terms of support, there are six sensations resulting from contact...

Alexander Berzin describes this mental factors as feeling (tshor-ba, Skt. vedanā) some level of happiness. He states:[7]

When we hear the word “feeling” in a Buddhist context, it’s only referring to this: feeling some level of happy or unhappy, somewhere on the spectrum. So, on the basis of pleasant contacting awareness—it comes easily to mind—we feel happy. Happiness is: we would like it to continue. And, on the basis of unpleasant contacting awareness—it doesn’t come easily to the mind, we basically want to get rid of it—we feel unhappiness. “Unhappiness” is the same word as “suffering” (mi-bde-ba, Skt. duhkha). Unhappiness is: I don’t want to continue this; I want to be parted from this.And neutral contacting awareness. We feel neutral about it—neither want to continue it nor to discontinue it...

Relation to "emotions"

Vedanā is the distinct valence or "hedonic tone" of emotional psychology, neurologically identified and isolated.

Contemporary teachers Bhikkhu Bodhi and Chögyam Trungpa Rinpoche clarify the relationship between vedanā (often translated as "feelings") and Western notions of "emotions."

Bhikkhu Bodhi writes:

"The Pali word vedanā does not signify emotion (which appears to be a complex phenomenon involving a variety of concomitant mental factors), but the bare affective quality of an experience, which may be either pleasant, painful or neutral."[3]

Chögyam Trungpa Rinpoche writes:

"In case [i.e. within the Buddhist teachings] 'feeling' is not quite our ordinary notion of feeling. It is not the feeling we take so seriously as, for instance, when we say, 'He hurt my feelings.' This kind of feeling that we take so seriously belongs to the fourth and fifth skandhas of concept and consciousness."[8]