For buyers and sellers this can sometimes be a confusing subject, but once you understand the differences you will be more comfortable.
An assessment is a value placed on a property by the Commissioner of Revenue's office in each locality. It is the dollar value that the property will be taxed on. Some localities assess every two years, some assess yearly. Either way, the tax rate passed by the Board of Supervisors or City Council is multiplied by the assessed value to arrive at your tax bill. Assessments are supposed to approximate market value and often do, but market changes between assessments and lag time from assessment to assessment can mean that sometimes an assessment is out of line with the current market.
An appraisal is a value a lender places on a property. Their purpose is to protect their investment (loan on the property). Different types of loans have various loan to value ratios. For example, an 80% loan to value ratio would allow a bank to loan $160,000 on a $200,000 property. A 95% loan to value ratio would allow a $95,000 loan on a $100,000 property. The lender wants to know that they have loaned less than the property value so if something goes wrong they have a reasonable chance of recovering their money.
Market value is what an able and willing buyer will pay for a particular property.
Sometimes these three things are very close in value. Sometimes they are not. If a property is assessed for $300,000 but a bank will only loan $175,000 they both can't be right. A person wanting to purchase this property would need to have enough cash to make up the difference in the offer price and the loan amount. This is an extreme example intended to demonstrate the practical differences in these values and the disparity is usually nowhere near the one shown here.
As a general rule, assessments tend to be high because local governments collect more tax money that way. Appraisals tend to be low because lenders want the extra margin of safety. Remember that in the final analysis, a property is worth what a willing and able buyer will pay.