A variance can be described as official permission from the city to allow something to be constructed or put in use within certain parameters that otherwise does not meet the standards of the City Code. For example, an applicant who wants to build an attached garage within required setbacks would need to request a variance from the Zoning Board of Adjustment (ZBA) and in their application explain why they are making this request. The ZBA reviews this application and makes a decision.
An exception is similar to a variance, but focused specifically on signs, parking, fences, telecommunications towers, and accessory uses. Like a variance, an applicant would need to request an exception from the ZBA explaining the reason for the request, and the ZBA would make a decision on the application.
More information on variances and exceptions, including the full description of application requirements and procedures can be found online.