What is the relationship between Thoreau and disobedience?

Thoreau may not have used the word "disobedience" with any frequency, but he did advocate for disobedience to unjust laws. He felt that we have a moral obligation to break the law when the law is unjust or morally wrong, such as when it supports the institution of slavery, one of Thoreau's major criticisms of the government. He refused to pay his poll tax, a tax levied on every citizen regardless of income, because he did not want his money to go to support slavery or the unjust Mexican war. For this crime, he was put in jail (though only for one night, because someone paid his tax for him—an action that really angered him). Thoreau believed that when the laws are not just, the only place for a just person is in jail because a just person must and should break those laws. In this way, he establishes his relationship to disobedience even without drumming the word again and again.

I'd say that Thoreau considers "disobedience" a kind of civic duty. This means more than simply disobeying laws; for Thoreau, government stands in opposition to the individual, and moral authority stems from individual conscience, not a voting majority. "Disobedience" therefore comes to embody a way of life that is morally disengaged from politics. In this way, it is possible to see his sojourn at Walden Pond as a form of "disobedience" in that Thoreau made a conscious choice to separate himself from a society that he viewed as corrupt. In the same way, his refusal to pay taxes, recounted in his essay "Civil Disobedience," is another example of how following one's conscience can lead one to a kind of "moral freedom," even though the result of his "disobedience" meant that he lost his physical freedom—he was locked in jail for a night.

Ironically, even though popular culture now firmly associates Henry David Thoreau with the concept of “civil disobedience,” the truth is that the author never used this phrase in his famous essay. Nor can it be found in any of his other writings.
Thoreau first presented his opinions on the subject in two lectures that he read in his hometown of Concord, Massachusetts, in early 1848. The first was called “Relation of Individual to State,” and the second one was “Rights and Duties of Individual.” When his resulting essay was published by Elizabeth Peabody in her single issue of Aesthetic Papers in 1849, it was under the title of “Resistance to Civil Government.” By the time it was published in book form – as contained in the posthumous volume, A Yankee in Canada, with Anti-Slavery and Reform Papers, in 1866 -- the title had been shortened and changed to “Civil Disobedience.” Henry Thoreau likely never considered using this phrase himself to describe the stand he took by spending one night in the Middlesex jail.
In fact, the only passage in the essay that includes the word “disobedience” appears in the twenty-third paragraph:

"No: until I want the protection of Massachusetts to be extended to me in some distant Southern port, where my liberty is endangered, or until I am bent solely on building up an estate at home by peaceful enterprise, I can afford to refuse allegiance to Massachusetts, and her right to my property and life. It costs me less in every sense to incur the penalty of disobedience to the State than it would to obey. I should feel as if I were worth less in that case."

Here he says that in most instances, he’s willing to pay the financial or moral costs to disobey the laws he believes to be unjust. His personal price would be higher, to do otherwise.