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.

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.

What is civilization in the mind of Huck in The Adventures of Huckleberry Finn?

To Huck, civilization represents repression and lack of autonomy. It also represent the rule of women. To Huck, it is cluttered with rules and rituals that don't make much sense. As he puts it, about living with the Widow Douglas:

The widow rung a bell for supper, and you had to come to time. When you got to the table you couldn't go right to eating, but you had to wait for the widow to tuck down her head and grumble a little over the victuals, though there warn't really anything the matter with them.

On the river, in contrast, Huck experiences a sense of freedom. It's not that he lives a disordered life, but that he is in charge and can follow a routine that makes sense to him. As he describes it:

Here is the way we put in the time. It was a monstrous big river down there—sometimes a mile and a half wide; we run nights, and laid up and hid daytimes; soon as night was most gone we stopped navigating and tied up . . . Then we set out the lines. Next we slid into the river and had a swim, so as to freshen up and cool off; then we set down on the sandy bottom where the water was about knee deep, and watched the daylight come.

On the river, Huck no longer has to live by other people's rules and expectations. He is his own boss.

But more profoundly, civilization represents a moral quandary for Huck. He has innocently imbibed its "morality," such as that it is a sin to help a slave escape to freedom. On the river, living outside of the constraints of civilization, he is able to develop a wider moral vision. The morality of his civilization condemns him and makes him feel guilty: he worries he will roast in hell for helping Jim. While we can laugh with and at his chafing under table manners and such conventions, we feel for the way the larger evils that are domesticated in the civilization he knows cramp the very possibility of the kind of expansive, humane relationship he is able to develop with Jim in the liminal space of the river. It is this deep sense of relationship we tend to respond to as readers, and we can understand the value of stepping outside of the boundaries of the known (i.e., the civilized) and into a world that allows our souls to expand. 

What is civilization in the mind of Huck in The Adventures of Huckleberry Finn?

Huck considers civilization hypocritical.  It means that people tell him what to do, and what they say is right and wrong is not necessarily what he considers right and wrong.  It means uncomfortable clothes and rules.  It is the opposite of freedom.

At the end of the book, Huck comments on why he left.

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 sivilize me, and I can't stand it. I been there before. (ch 43)

This is a common motif throughout the book—the contrast between civilization and humanity.  If civilization means confining clothes, boring school, and stifling religion, Huck can do without it.

Although Huck cannot stand his repulsive and abusive father, he still prefers being able to fish, eat stew, wear comfortable clothes, and sleep outside to being "cramped up" with the widow (ch 6).  Huck has grown up wild, and since most people cannot explain why the rules of civilization exist, Huck sees no reason to follow them.