As mentioned by other members here: there is biting, and then there is preening. But there is also playtime and curiosity.
My two previous hens, Nutmeg and Luna are a good example of one who loves playtime and the other who bites and makes you bleed, but, it wasn't always this way. Let me explain.
Nutmeg and Luna, in my opinion, were homed in a neglectful environment. They needed to relearn that humans are not scary and that hands will not hurt. Nutmeg came around much much sooner than Luna and we developed a special bond. She loved to play with my fingers and land on my hand or near me just to play attack my fingers. Then she would fly off and do her thing, and then come back for more. They were the gentlest of nibbles and bites and she loved talking to me in her little trill. She was very dominant and had a heckuva lot of moxie. We had a hoot! and I was thoroughly entertained by her antics.
Luna was VERY different. It took a year to work with her to have her feel comfortable with me and my partner as flock mates. You couldn't touch her. She went into complete and utter flip out mood and if she managed to get her beak on you, she would mark you and make you bleed. She had a tremendously HARD bite for a small bird. The first year was hard, but she slowly progressed into feeling more comfortable with her surroundings and humans. She became very affectionate when SHE thought it was okay: nibbling my elbows, hopping on my shoulder, letting me stroke her chest area, preening my ear or giving me a kiss (the smallest of occasions) on my cheek and relearning what 'step up' is and flying and landing on my hand or arm.
Even with encouragement, Luna was still deeply cautious, which made me think that she was very abused or possibly attacked by another larger bird in her previous home. Still, I respected her boundaries and was very aware of them. My training with my birds is slightly different in that sometimes I had to push those boundaries just a little bit, but not to cause harm or stress, just to encourage a continued connection so that the bird was familiar with imprinting in the routine. You can do that more slowly, as well, but it really depends on the birds comfort zone. Luna, unfortunately, never really had a comfort zone, and would not be fully at ease with humans and more aggressive birds, which is why Nutmeg and Luna needed separate cages. Nutmeg also became very jealous if other birds received more attention than her so I had to be careful and encourage share time. I made a point of giving all my birds as much equal attention as possible. But Nutmeg was her own birdie, full of moxie that one.
In the end, not all biting or nibbling is bad. Budgies are super curious about most everything and really enjoy showing us just what they 'like' and what they can do. You will know what a BITE is. If it is gentle nibbling or biting, it is a move toward getting to know you a bit better and showing you other affectionate ways of saying "Hey, you're okay."
Just one word, regardless of the bite, do not pull away. They need to understand, no matter how painful that bite may be, that it doesn't affect you. Pulling away when they are biting hard allows them to condition to knowing that they have successfully subdued you. Of course, there is a time to engage and a time to take a break -- if your budgie is gentle one minute and changes their bite to be more aggressive, then it is time to give them a break and let them "cool their jet skis". Keeping them engaged, however, will help them to know that you are not going to hurt them, and that this thing (your hand) is safe.
Enjoy your budgie -- bites and all!