The Meaning of Vedanā in Buddhism
Vedanā is an essential concept in Buddhism. It made quite a number of prominent lists, including the Five Aggregates (khandhas) and the Four Foundations of Mindfulness (satipaṭṭhānas), and it would be difficult to overstate its importance for meditation practice.
Vedanā has been translated as "feeling" or "sensation". According to the early Buddhist discourses (e.g. Datthabba Sutta; SN 36.5), there are three kinds of feelings: pleasant ("sukhā vedanā"), painful ("dukkhā vedanā"), and neither pleasant nor painful ("adukkhamasukhā vedanā"). The latter is also sometimes translated as "neutral" feeling.
Most experiences come as either pleasant or painful, and when they do not, then surely "neither pleasant nor painful" covers the rest. Does that mean that any and all experiences are vedanā? No, it does not. It just means that vedanā is constantly present, and we might be having trouble telling it apart from the rest of experience.
Vedanā and the Body
Vedanā is the second of the four satipaṭṭhānas (Satipatthana Sutta; MN 10). The first satipaṭṭhāna is the body (kayā). Translating vedanā as "feeling" or "sensation" can cause some confusion because we know the body by way of feeling/sensation. If I have a pain in the knee, does that belong to kayā, vedanā, both, or neither?
Neurobiology has uncovered a clear distinction between two types of feelings. The first is about the feelings of the body moving through space (proprioception) and discriminative touch. It begins with mechanoreceptors e.g. in striated muscle tissue and skin, which are relatively large neurons that become active when these tissues are stretched or contracted. Via large-diameter, fast conducting fibers, they signal a brain area called somatosensory cortex, which integrates that information and sends it to another area called insular cortex or insula for short. The second type of feeling, called interoception, begins with receptors that are sensitive e.g. to distension of smooth muscle tissue, tissue damage, chemicals, temperature, and affective (tender) touch of the skin. These neurons provide information about the condition or state of the body. They project via small-diameter, slow conducting fibers not just to somatosensory cortex, but also directly to the insula and a few other brain areas. In the insula, then, all the signals from the body become integrated to form the whole of the feeling body as we know it. I got all of that from neuroanatomist A.D. (Bud) Craig's book "How Do You Feel? An Interoceptive Moment with Your Neurobiological Self", if you're interested in reading more.
Kayā refers to the first type of feeling, informing about how the body moves and what it touches. Vedanā refers to interoception, informing us about the condition of the body. A pain in the knee belongs to vedanā, whereas e.g. the feeling of moving an arm belongs to kayā. However, if the arm muscles are sore, the soreness belongs to vedanā. Interoceptive feelings (vedanā) can be pleasant, painful or neutral, but this categorization scheme does not apply to non-interoceptive feelings (kayā).
Vedanā and Hedonic Valence
Some believe vedanā referred to the pleasantness and unpleasantness (hedonic valence, feeling tone) of a given experience. But if that were true, the concept of neither-painful-nor-pleasant (neutral) feeling would be void; it would not refer to anything, for one cannot sense "neutrality itself", only that something - a feeling (vedanā) - is neutral. Therefore, the concept of vedanā must have an existence independent from its hedonic valence.
Vedanā and Emotion
What is the relationship between vedanā and emotions like hunger or love?
I like the idea that emotions are learned patterns of bodily changes caused by autonomic bodily activity in response to stimuli (see e.g. the "James-Lange theory of emotion" and the "somatic marker hypothesis"). Basically, something happens in your environment (or in the body itself) that causes a response by your autonomic nervous system, that, in turn, leads to changes in the body (e.g. an increase in heart rate). These changes stimulate interoceptive receptors in affected parts of the body, which transmit signals to insular cortex. The resulting feelings receive a valence, which is positive (pleasant) if the bodily changes are supposedly evolutionarily helpful, negative (unpleasant/painful) if they might be evolutionarily harmful, and neutral if evolution couldn't care less. If a pattern of feelings is experienced often enough, we learn to recognize it, and maybe give it a name like "hunger" or "love".
Given this model of emotion, vedanā would be the interoceptive basis of emotion, i.e. what the body feels like during an emotional experience.
Vedanā in Meditation
Meditating on vedanā means constantly observing one's changing interoceptive feelings. If a feeling is pleasant, one should remember that pleasure, if clung to, leads to desire, and desire leads to suffering. If it is painful, one should remember that pain, if one clings to painlessness, leads to aversion, and aversion leads to suffering. If it is neutral, one should remember that clinging to the status quo leads to suffering when things eventually change. See also the Pahāna Sutta (SN 36.3).