The way I see it, feelings are the lowest-level reaction the brain has to a stimulus. A stimulus can come from the senses or from thoughts, which are influenced by both feelings and the senses, so thinking about something can influence your feelings. It's a constant feedback loop which is partly influenced by the world around us, and partly by our conscious thoughts. All emotions have a biological reason to exist. Take fear for instance, if something terrifies you, you end up reacting to it much faster than if you took time to think about it rationally (fight or flight response). It makes biological sense that the brain would have some mechanism to ensure rapid response to danger. Joy and pleasure have the purpose of reinforcing a person's desire to do something they enjoyed the first time again. Love, in it's most general sense, results in the formation of the family unit and allows for a cohesive group (a tribe or community) to form, assuring survival of the species. But that's still mostly "why" and not "what." Emotional responses occur in the brain's limbic system, which is at the center, near the brain stem. It's hard to really say what an emotion is because, as far as I know, nobody really understands how the complex chemical reactions in the brain get turned into conscious experiences. Perhaps it is simply outside the scope of current human knowledge.
As far as being overcome by emotion goes, it sometimes makes sense that the brain would subdue reason for the sake of acting quickly and with conviction, or to share emotion with others via empathy (why its hard to stop crying). Emotions can also act as the fruits of the labor which is thinking by providing us with the motivation to do something after reaching a satisfying logical conclusion. Transmitting emotions is simply the lowest-level form of communication. It is done instinctively among social animals, like ourselves, and can be understood even across wildly different species. You probably clench your fists without consciously thinking about it if you get angry, and you might make an angry sound analogous to a dog's growl. I think some emotional responses can be likened to a computer program where certain sections of the code are run only when a variable increases past a certain value, with the variable being emotion. These sections of code would be like the "firmware" of our minds, Post too long. Click here to view the full text.