Excellent question, as I am trying to define good and evil (in the Biblical context) as truths within Archetypical stories, rather than defined in relation to an embodied God. So, I guess I am defining evil and sin as something like, 'intentional acts which hurt others or ourselves (physically or psychologically), and (to some extent) the dark emotions which tempt or provoke us to those actions.'
In other words, I am trying for a definition of sin that does not require a deity to define it.
Now, of course, in Tolkien's Arda (or mostly outside of Arda) there is a God, and a variety of Angels, but we don't have a very clear idea as to what Eru regards as sin.
The Christian definition of sin is also not very clear. St. Augustine defined it as, "a word, deed, or desire in opposition to the eternal law of God".
I think I would agree with that (especially if defining 'God' as symbolic of Human nature and the created world), but I don't think we know what the eternal law of God is exactly.
I am certainly not a Biblical literalist, nor fundamentalist. I think that the books of the Old Testament are a collection of stories told orally by generations of men, refined until they became archetypal stories that contain true things, then written down and edited and assembled by other wise men at various times, and certainly not dictated by God.
So, which statements attributed to God can be interpreted as 'the eternal law of God' is dubious. Is any given statement attributed to God advice? instruction? a wish? a hope? a command? a law? eternal? Very hard to say, and also contradictory at many points across the Old Testament.
The other Christian definition of sin that I am aware of is, 'an act of offense against God by despising His persons, and Biblical law, and by injuring others'. Which does not help much, as it comes back to the question of what is Biblical law?
Let's take an example from the story of Adam and Eve: 'And the Lord God commanded the man, "You are free to eat from any tree in the garden; but you must not eat from the tree of the knowledge of good and evil, for when you eat from it, you will certainly die."' Is that a law, or a command, or an instruction, or just advice (if you eat it you will certainly die). Is it eternal?
I read it as an archetypal story explaining that when men evolved consciousness, they gained a knowledge of good and evil. When they became conscious, they also became aware (unlike, or differently from animals) that they would certainly die. I don't need any God, or any disobedience, to read that 'truth' into the story.
So, then, I wonder, 'what is evil'? Well, once conscious, Men can plan to hurt or increase the suffering of others to gain future objectives. That is evil. Men can also hurt themselves or others out of despair, rage, envy, nihilism, due to their 'sinful' perspective on the prospect of a short and suffering life, ending in inevitable death. That is evil.
I regard the Adam and Eve story as 'true' in describing the 'fall' of Man into consciousness and the knowledge of good and evil. Unlike the church, however, I don't regard the 'fall' of Man as 'original sin'. Man had no volition in acquiring consciousness. Evolution did it. The consequence of gaining a knowledge of good and evil was just an inevitability. Man has to learn to live with it. God is not necessary to explain it.
I hope that is at all clear? Defining metaphysical concepts like sin and evil (especially if eschewing the easy appeal to divine authority) is difficult. Stories do it better than articulation.