Zoning codes apply the same rules to every lot in a district, but not every lot is ordinary. A variance lets a board relax a rule for one property when its shape, size, slope or other condition makes the rule unusually hard to meet. Most variances are dimensional: a smaller setback, more height, more lot coverage, fewer parking spaces.
In most of the United States, variances are decided by a board of zoning appeals, called a board of adjustment or a zoning hearing board in some states, at a public hearing with notice to neighbors. The applicant must show the case meets a legal test, not just that the change would be convenient.
The test is set by state law and the local code, and it is similar in most places: strict application of the rule would cause an unnecessary hardship; the hardship comes from conditions peculiar to the property, not from the owner's own actions; and the variance would not harm neighbors or the character of the area. Many states do not allow a use variance, which would permit a use the district forbids; that takes a rezoning.
A board can attach conditions to a variance, and a neighbor or the applicant can usually appeal the decision to court within a short window. In South Carolina, for example, as in Charleston, variances go to the board of zoning appeals under the state's planning enabling act.