I would say that at the inception of an emotional episode there is some event, either in the environment or the person's own mind, that undergoes cognitive appraisal aimed at estimating the event's likely impact on the person's welfare. The appraisal might be the quick-and-dirty, unconscious or barely conscious, kind of assessment that amygdalas are good at. Alternatively, or in addition, it might involve prefrontal-cortical structures that work more slowly but are able to use a lot more of the information at the person's disposal.
The outcome of the appraisal, if it indicates the need for some action, is passed to limbic structures and the peripheral nervous system, both autonomic and voluntary. The nature of the indicated action -- approach, attack, avoidance, or whatever -- determines the specific content and pattern of physiological activation. The activation can ordinarily be felt proprioceptively, giving rise to the subjective experience of having a "feeling."
The above is a stripped-down, linear, formulation of my understanding of what happens in the course of an emotional episode. In practice, there are feedback loops linking "downstream" phases of the process back to "upstream" phases, resulting in dynamic structures that can become quite complex.
For more complete and authoritative accounts of these processes than I am capable of giving, I would recommend the
following recent books:
LeDoux, Joseph. The Emotional Brain: The Mysterious Underpinnings of Emotional Life. Simon & Schuster, 1996.
Damasio, Antonio R. Descartes' Error: Emotion, Reason and the Human Brain. Avon Books, 1994.