Morality is the concept of doing the right thing. Morals are basic guidelines for living. Many people have written about ways of choosing what the right thing is. Some believe that there is an objective (true even if you do not agree with it) morality, often thinking it was set out by God or another similar being; others think that morality is subjective (that what is good or bad is your choice).
It can also be a lesson that someone learns in a book or story. It is usually one line at the end of the story ("The moral of this story is....."). Fables are stories with a moral.
The opposites of "moral" is "immoral", meaning the wrong thing to do. "Amoral" means something that is not related to morality, or has no morally good or bad aspect to it. It is the same as "morally neutral".