Can someone please explain the connection between electrostatic potential and Laplace's equation? And what does that have to do with harmonic functions?
This is a real broad brush question. A decent answer would take up several chapters of a book. I can say this with confidence because I have studied the chapters of such books.
In my opinion you're best advised to consult any decent undergraduate textbook on electro-magnetism and study the salient chapters at your leisure.
eg. Lorraine and Corson - Electromagnetic Fields and Waves, Jackson -Classical Electrodynamics.
In a nutshell:
1. In regions of space where there is no charge density the electrostatic potential satisfies Laplaces equation: $\displaystyle \nabla^2 V = 0$ where V is the electrostatic potential.
2. Any real function with continuous second partial derivatives which satisfies Laplace's equation is called a harmonic function.
Background knowledge, derivation, details and examples (and answers to the million and one questions the nutshell has prompted) are what constitute the above-mentioned chapters.
Just curious by the way ...... how do you happen to be in the unfortunate circumstance of this question being urgent homework?