Question

How do messy galaxies turn into organized spirals or pretty galaxies?

Answer 1

It is nearly all due to gravity alone at first. Mass, for reasons we still do not know, attracts other mass too it and slowly , as things gravitate together, structure emerges from pressure and heat with stars forming heavy things eventually

Answer 2

Gravity is a big part of galaxy formation as it acts at very large distances, but the specifics of how different types of galaxies form vary. Here's a place to start answering your question: http://en.wikipedia.org/wiki/Galaxy_formation_and_evolution