What is anger? It is an emotion. What are emotions? They are, or include affective states.
Anger is an affective state in which the person concerned is unhappy in response to a situation in which they feel they have been treated unfairly and in which they strongly desire to change the situation. However, there is more to it than that... anger is more than just a desire to change something in response to a disappointment.
When angry, a creature wants to do more than just change its situation; it wants to make things worse for those it regards as responsible for the situation -- revenge.
But that, too, is not all of anger... although anger is normally directed by the subject towards the particular objects that they see as being responsible for their perceived unfair treatment, there is also generalized anger (directed at everything that comes in handy).
This is part of what fascinates me about anger (without having much effect on my own anger): that when one thing angers us, we become angry in general, and much more easily irritated by other things not going just as we would like.
In cases of continuing intense directed anger, the subject may develop hatred for the object... although hatred may appear other than as a consequent of anger. In hatred, the object is regarded as inherently bad, and the subject has very negligible regard for the needs of the object, either wishing for it to be removed from the subject's world, or for it to be made to suffer indefinitely -- way beyond any normal level of retribution.
And yet, this is still not all of anger... that fact that we can feel angry is a useful pointer to our consciousness; it seems to me to be not simply a change in state that could be modelled mathematically (although I'm sure it includes something which we could model reasonably well) but is also something that we feel... something that goes down into the soul, whatever that is.
It seems to me that there is more than one kind of anger, determined by its origin:
Disgruntled thoughts lead to angry feelings, and angry feelings encourage disgruntled thoughts, and bias our understanding of what we see. In terms of our mental terrain, we get stuck in a red valley, and it's difficult to get back to the high ground of unconstrained free will, in which we can make choices without this bias.
The Book of Proverbs has much to say on the foolishness of decisions made during anger.
Cognitive Therapy is a modern framework for breaking the cycles of thought and feeling that often trap us.
In my notes in Christian Cognitive Therapy, these are combined.
Anger may motivate us to get something done that we wouldn't otherwise have done, but it tends to produce irrational decisions and opinions. It would be better to be able to acheive the same level of motivation while remaining calm.