A tax assessment is a government assigned figure that property taxes are based on. In some "ad valorem" states an assessment is related to market value and it is determined or established by analyzing comparable sales and cost data. In other states, like Nevada, assessed values are based entirely on cost estimates. An assessed value is usually a portion of a property's "taxable value" which is the Assessor's estimate of full value. In a large metropolitan area an Assessor often uses "mass appraisal" techniques / computer models to assign taxable values to each property. Assessed values, taken from the taxable value estimated, are multiplied by tax rates to determine their taxes.
Appraisals are market value opinions formed via the appraisal process by licensed appraisers. Appraisals utilize the three accepted appraisal methodologies (the cost approach, the income approach and the sales comparison approach). Appraisals are usually completed on a single property with all of its physical, locational and economic characteristics considered, and the appraisers are usually private party individuals and not government employees (although some appraisers do appraise real property for the government).
If you are located in an "ad valorem" state you can often file an appeal with the Assessor if you believe that the assessed value / taxable value assigned to your property is not representative of the it's market value. Some states require that in order to mount a challenge to the assessed value / taxable value established by the Assessor the appeal must be submitted with an appraisal report. In Nevada tax appeal cases require an appraisal and the appraiser must attend the hearing to defend his work.
Assessed values and taxable values are usually reported to property owners annually, and they can change over time. The assessed / taxable value for any particular property may be reasonable, close to market value, one year and then be off another. It's not like there are wide swings year-to-year, but it is possible that your assessment moved up while the property value moved down. Since appraisals are specific to a single property and not based on "mass appraisal" techniques they are generally more accurate than assessments.