Assessor as evaluator
An assessor is an expert who calculates the value of property. The value calculated by the assessor is then used as the basis for determining the amounts to be paid or assessed for tax or insurance purposes.
Assessor as government official
In local government in the United States, an assessor is an appointed or elected official charged with determining the value of each taxable property in a county, municipality, or township; this information is then used by the locality to determine the necessary rate of taxation to support the community's annual budget. (This is a specialization of the previous sense; a person who performs similar work for a private employer is more often called an appraiser or, specifically in the insurance industry, an adjuster.)