What are the different types of value?  How can there be different values for the same item?
Personal property appraisals are written for a variety of different types of value levels depending on the intended use of the appraisal.  These types of value can include fair market value, retail replacement value, marketable cash value, actual cash value, liquidation value, and salvage value.  Different intended uses for an appraisal correspond to different types of value in varying levels of the marketplace, such as equitable distribution with divorce, forced sale with bankruptcy, charitable contributions, or an insurance damage claim.  The appropriate marketplace for a value level can also depend on the specific terms of an insurance policy or legal court settlement.  Therefore, depending on the intended use of the appraisal (which is the reason why you need to have an appraisal written), the exact same item could have multiple different appraised values if the appraisal had different intended uses and thus used different value types.

The most commonly encountered types of value in appraisal assignments are retail replacement value and fair market value.  In general, retail replacement value is the type of value used for appraisals for insurance purposes (although this can be different depending on your specific insurance policy), and fair market value is used for appraisals for federal tax functions.  

The International Society of Appraisers defines replacement cost as "the cost to replace an item with another having similar qualities within a reasonable amount of time in the relevant marketplace.  It takes into consideration possible costs incurred by engaging an interior designer, shipping, and the costs associated with purchasing from a specific source."  Often replacement cost will be the higher than the other types of value levels as it reflects the intended use of providing sufficient insurance protection for you if you have to replace your appraised item in the future in the event of a loss.

Fair market value is the type of value required for all appraisals written for federal tax functions.  The Internal Revenue Service defines fair market value as "the price that property would sell for on the open market.  It is the price that would be agreed on between a willing buyer and a willing seller, with neither being required to act, and both having reasonable knowledge of the relevant facts" (IRS Publication 561:  Determining the Value of Donated Property, available here)