In addition to what Abigal said....
When a bird bites out of fear, the bird would be leaning away and trying to avoid the situation. They bite as a 'last resort' and would prefer to avoid the situation rather than to confront it. If you find that a bird is biting from fear then you need to back off and give the bird time to settle. Then work *slowly* with the bird and reward the bird for showing interest in you and treats from you. This basically means *DON'T CHASE THE BIRD!*. I know that many owners when they get a new bird they instantly want to interact with the bird, but some birds need time to settle in before they are ready to have any physical contact with a person, and if that's the case you just need to have patience and take things slow.
When a bird is biting out of curiosity, they show no behaviors of being afraid. In fact, it looks like they are exploring. Parrots don't have hands and they explore a lot using their beaks. They will often 'test' things using their beaks and may apply different pressure to what they are checking out. You don't want to "ignore" this behavior, and likewise, you don't want to discourage it, either! If you ignore the behavior, the bird mite bite harder and harder until something happens. If something happens and it ends up being a scary experience for the bird, you can set taming progress back with the bird. If you discourage the behavior, then the bird may not learn to investigate new toys or objects. You actually *WANT* them to explore! But you need to teach them what is and is not appropriate to chew. If he shows signs of being interested in chewing on you, then give him something else to chew on instead! Try plastic/wooden beads, pieces of vegetable tanned leather, bottle caps, maybe some food/treats twisted up inside a paper cupcake holder, or other things of potential interest to the birds! Heck, you could even get some plastic straws, cut them up, align the straws together and tie them in the center. Once you do that, you can then stuff the ends of the straws with food and you've got an instant foraging toy!
When you see an undesired behavior, instead of asking yourself "How do I stop this behavior?", instead ask yourself "What do I want my bird to be doing instead of this behavior?"
The difference is, that instead of teaching your bird to "do nothing", you teach them behaviors *to do* instead of behaviors you don't want them to do. As you continue to reward the bird for desired behaviors, they are more likely to do the desired behaviors over undesired behaviors.
Here, this post from Barbara may help!
Good Bird Inc Parrot Training Talk: How to Make Friends with a Parrot
