Atomic stability

Okay, We all know that protons and electrons are oppositely charged particles and as such should be drawn towards each other but in the atom the electron maintains a respectful distance from the core. Could someone please explain how this process works (the less math the better), what is it that keeps them from coaleseing?

You know how the moon and the earth are positively attracted to one another via gravity?   It's the exact same concept.

Of course this is a simplified classical model of the atom, but unless you're getting into quantum mechanics it's a useful model.  It can even be used, with a small modification, to get a good approximation of orbital radii.