From the very outset of the play, civilization among the boys is under attack.  It is represented in many ways by Ralph, by the conch, by the choir boys in uniform marching down the beach, and by Piggy and his desire to think things through and be rational.

As the boys get hungrier and hungrier and begin to descend into a more animalistic state, the civilization they've established quickly starts to unravel as do its representatives in Ralph and Piggy and the authority of the conch.

Eventually the entire idea of a civilized society is destroyed as the boys become a tribe of animals, driven to hunt and to kill, as they kill Simon, then Piggy, then bring about the destruction (at least symbolically) of the island by starting a fire that consumes much of it and perhaps many of the boys as well.

What is the significance of the theme of civilization in Lord of the Flies?

One of the other things about the theme of civilization not mentioned by the previous and very thorough post, is the way that it drove Golding to write the novel in the first place.  Golding felt that British boys were just as likely to forgo the trappings of civilization as anyone else, a slightly different view that that expressed in Byrne's "Coral Island."

So Piggy and Ralph represent this idea of civilzation being important but Jack and the others demonstrate that the evil within these boys is real and that they can give in to it as soon as the rules and trappings of society disappear, of course Roger being the example of a sociopath who goes further down that road that anyone.

What is the significance of the theme of civilization in Lord of the Flies?

The theme of civilization, or more precisely, the various aspects of civilization, is central to "Lord of the Flies".  The question that the novel asks is, "What happens to people when society's rules are no longer applied?"  In the story, we see good v. evil in the form of the good, civilized tribe consisting primarily of Ralph, Piggy, and Simon along with some younger boys, and in the form of the evil, savage tribe led by Jack, Roger, Maurice and an increasing number of the other boys.  While Ralph's tribe tries to maintain civility by following the old rules, Jack's tribe abandons the ties of civilization and Jack creates a dictatorship keeping most of the other boys in line by keeping them fearful of him and his henchmen.  Piggy, especially, is the boy who sees that the rules must be followed or chaos would ensue and he is killed for this observation in chapter 11 when he pleads with Jack and his tribe to listen to reason.  Jack's group has lost their moral values that they had when they were members of society.