Huck, who phonetically writes "civilize" as "sivilize," is accustomed to living on his own without the rules of civilization to worry about. He is not worried about manners, about appearance, or about conforming to society. At the beginning of the book, he has just been made rich by the discovery of money at the end of the previous book, and so he is adopted by the Widow Douglas:
The Widow Douglas she took me for her son, and allowed she would sivilize me; but it was rough living in the house all the time, considering how dismal regular and decent the widow was in all her ways...
(Twain, The Adventures of Huckleberry Finn, gutenberg.org)
To Huck, civilization is far too restrictive to live in; he can't do what he wants without some sort of rule defining his behavior. He decides to escape, ends up going back, and is constantly at-odds with society. At the end of the book, he again goes on the run, refusing to be "sivilized" by Tom's Aunt Sally; he claims to have been in civilization long enough to know that he can't stand it. This shows Huck's individualism and his general isolationist tendencies; he doesn't need other people to be happy, and so he doesn't think that he needs their rules either.
For Huck Finn, "civilization" represents the mores of the slave-owning society that he thinks he should follow but that he can't actually follow. Huck is hard on himself for not being able to abide by the laws of the slave society he comes from. When he nears freedom with Jim, the escaped slave he is trying to bring north, Huck hears Jim speaking about how close he is to freedom. Huck thinks,
Well, I can tell you it made me all over trembly and feverish too, to hear him, because I begun to get it through my head that he WAS most free--and who was to blame for it? Why, ME. I couldn't get that out of my conscience (page 98).
Huck is doing the moral thing by trying to remove Jim from the inhumane horrors of slavery, but because the civilization from which he comes is hypocritical and practices slavery while referring to themselves as moral and religious, Huck feels he is doing the wrong thing.
Huck has the integrity to stick to what he feels is right, even though what he does goes against the morals of the society in which he lives. In the end, the disparity between what he feels is right and what society thinks is right makes him eager to escape from civilization again. At the end of the book, he says: "But I reckon I got to light out for the Territory ahead of the rest, because Aunt Sally she's going to adopt me and silivize me, and I can't stand it" (page 327). Women in the novel, including Miss Watson, are constantly trying to force Huck to conform to the ways of the society, including quitting smoking and reading the Bible. Huck knows that he can't live among so-called civilized people because of the hypocrisy of their morals and their willingness to crush his spirit and boss him around. He knows he has to live in the west, which is not yet settled, so he can live the way he wants to live.