What does it mean when a bump occurs next to the pierced hole in your nose?

Well it's called a "Hypertrophic Reaction". It's a reaction to the cleaner you are using on your piercing, this can be facial scrubs, cleaners, soaps and what ever else you are using on your face in the area of the piercing. Flush the piercing out daily under running water while moving the jewellery to rinse the stuff out of the piercing. Don't use any cleaners or scrubs in the area of the piercing. The flushing daily will allow the piercing to settle down over the next few days.