What are emotions in the 21st century?

What is the relationship between emotions and the brain? Which brain structures are involved in the generation and regulation of emotions? Whence emotional upheaval? Whence emotional behavior?
These questions has now been well answered by 21st century neuroscience. Lisa Feldman Barrett studied this extensively and has summarized her work in
How Emotions Are Made by Lisa Feldman Barrett
She first found that what society and psychology call emotions are really two kinds of thing. First and foremost are the unconscious biological responses that are called affect, which are how each of us maintain all bodily systems across large and sudden changes in the external environment (AKA threats). Second are the memories we form of previous threats to our bodily integrity; memories that record both how we previously responded and how the responses played out.
Affect is mediated primarily by the autonomic nervous system (ANS), which includes the limbic system and extends into every part of the body. The limbic system and ANS work ceaselessly to adjust bodily processes such as heart rate, breathing rate, hormone levels, vascular tension, blood glucose levels, attentiveness, etc. They monitor all these systems and adjust them, trying to keep the body alive under changing circumstances. This is done by predicting how the body will respond in the next second or so, and uses ANS and central nervous system (CNS) to update the predictions and test them.
The limbic system consists primarily of the prefrontal cortex (PFC), amygdala, and hippocampus. It not only predicts what will happen inside the body; it also predicts what our external senses (primarily vision, hearing, and touch) will discover in the next second or so. It projects these sensory predictions down through the various types of sensory cortex to the thalamus, where the processed predictions meet our external sensations shortly after they arrive at our retina, ear, and skin. When the sensations don’t match the predictions, two things happen simultaneously: the prediction errors are pushed back up through sensory cortex to update the predictions, which can take hundreds of milliseconds, and the amygdala is notified directly of the mismatch occurence and the degree of mismatch, which takes only a few milliseconds. The report of a mismatch is included in the brain’s autonomic predictions.
The mismatched predictions, both internal (ANS) and external (sensory) are used by the limbic system to adjust our bodily functions. The adjustments can include increasing heart and breathing rates, releasing stress hormones (primarily cortisol and epinephrine), constricting blood vessels, redirecting attention, and dozens of other internal adjustments. These responses are called affective reactions, and they form the first line of our emotional responses.
The second part of an emotional response is mediated by the hippocampus and PFC. Whenever the limbic system senses a threat, internal or external, both start recalling memories of similar experiences, and it is these experiences that become our emotional behaviors. Actually, most emotional experiences are primed by memories leading up to the perceived threat, as well. The purpose of all this recall is access to previously learned survival strategies.
Even before the hippocampus can recall relevant episodic memories, which can take hundreds of milliseconds), the prefrontal cortex can recall responses to traumatic circumstances that trigger what might be called survival behaviors. These behaviors are essentially hard-wired responses (tens of milliseconds), and they can be made worse by myelinizations that both ensure faster response and limit plasticity.
Emotional behaviors are how we respond to affect. The noises we make. The apparent reflexive behaviors we exhibit. Our self-enforced silence. The crying. The hitting. The yelling. The over-reactions in our bodily functions. The aggression. The sustained attacks. The words we use to insult and badger. All are learned behaviors, very much different than the affects that induce them.
That is not to say that our emotional behaviors are conscious and intentional. They are learned and ingrained, so they appear to us as a continuum with the affects that trigger them. They are effectively habits that we have learned in order to survive.
While there is some evidence that we can learn to modulate our affective responses, we know that they are very hard to change, partially because they are entirely unconscious, partially because some of them are mediated by traumatic learning.
Our emotional behaviors, on the other hand, are more amenable to change. Understanding that there is a neurological difference between what we experience and how we respond to it can affect our emotional lives.
We have always known that habits are hard to change. By knowing about the boundary between emotional experience (affect) and emotional behavior, we can work to change behaviors without being existentially frustrated by the intractable autonomic responses that trigger them.
Another reason this knowledge should give us hope is that emotional behaviors such as lashing out physically often induce autonomic changes (e.g., heart rate) that further trigger changes in affect. This feedback loop is a major source of emotional difficulties.
Dr. Barrett’s discoveries have only been published this century. They have been entirely adopted by the neuroscience community, and their implications are slowly being adopted by the psychological and psychiatric communities.