The world in miniature

In school, we learn that the world is made up of atoms, elements and bits and pieces of matter. They forget to mention that those pieces of matter are mostly made up of empty space with a few electrons zinging around in between. I remember asking my chemistry teacher why don't electrons go flying away from the atom. If i recall correctly her answer involved something along the lines of it being attracted to the proton at the centre of the atom, which is positively charged. They teach you that opposites attract at school, at least at the level of atoms. But then I always wondered, what holds the neutrons in place, they're neutral, aren't they? Do they really just sit there and do nothing? What is really going on at the level of atoms, and what did they omit to tell me in school? Perhaps Wikipedia will hold some insight for me.