Meaning is that abstract, fuzzy thing in your head that a word or a phrase represents. It includes what the word denotes and what the word connotes, but it also carries associations in memory, the context in which it occurs at the present and in which it had occurred in the past, class , regionality, ethnicity -- a whole panoply of things that, in the end, prevent most words from ever meaning precisely the same thing to two different people.
A definition of a word is an explicit statement in other terms that is intended to capture the meaning of the word. In a formal, sense one should be able to replace the word with the contents of its definition, but often the complexity of meaning of an individual word is too great to be captured in a short sentence.
Ordinary language is, at best, a tool that allows us to convey a very good approximation of what we mean to others. (We'll leave the narrowly-defined and deliberately precise professional and technical vocabularies aside for the moment.) Obviously, there isn't going to be a lot of communication going on if we can't reach some sort of agreement on the broad meaning of words. If I use red to talk about a small furry creature that you would call Thursday -- well, that's how wars get started. So we agree to call it a hamster for no good reason other than that we can agree on the word. We can express that broad agreement in a definition, an alternate word or phrase that means approximately the same thing to both of us. Still, the word hamster means something completely different to my sister (who loved her pet dearly and was devasted by its passing) than it does to me (who was annoyed by the racket it made, disgusted by the droppings it left everywhere, and who had to tear the ductwork apart to retrieve the thing after it made its way through the cold air return register), even if we agree on the definition of the word, and therefore the looser sense of the word meaning.
A dictionary is a collection of those alternate words and descriptive phrases, those definitions that we've agreed upon. A very good dictionary may define a word well enough that you begin to get a sense of its meaning, but ultimately the word will mean whatever that abstract, fuzzy thing in your head that it points to tells you it means.