Some uses fit a district in principle but need a closer look on a particular site: a school or church in a residential area, a gas station on a commercial corridor, a larger home business. Many codes list these as special exceptions. The use is anticipated by the code; what is decided is whether this site and this design meet the standards the code sets for it.
In many states the board of zoning appeals hears special exceptions at a public hearing with notice to neighbors. The standards are usually written in the code, such as traffic, parking, buffering, hours and compatibility with the surroundings, and the applicant carries the burden of showing they are met. The board can attach conditions to an approval.
The names vary. What one state calls a special exception, another calls a special use permit or a conditional use permit, and the deciding body may be the board of zoning appeals, the planning commission or the governing body. The idea is the same: an allowed use that needs a hearing.
A special exception is different from a variance. A variance relaxes a rule because of a hardship peculiar to the property; a special exception grants a use the code already provides for, once its standards are shown to be met. In Charleston County, South Carolina, for example, the zoning use table marks some housing types as allowed only by special exception.