The county determines your assessed value and an appraiser determines your market value.
It is also good to know what percent of valuation your city or town uses to determine tax rate. You can find this out by looking at your tax bill or contacting your town or city assessors office.
Depending on what state you live in your assessed value can change when you do home improvements. When you get permitted they re-assess your home.
The assessed value is nearly always out of date by the time you get it. Each state has different laws determing what date the estimate of value is to be determined for.
A realtor will tell you the Fair Market Value is the highest price agreed upon. Most underwriters and appraisers define it as the most likely price to be agreed upon. That's why underwriters insist upon several comparables in the appraisal report, and don't usually just go by the sale price.
In California, the tax collector may not change the assessed value of a residential property until the property is transferred. This is a protection known as Proposition 13 protection, named for a public referendum that initiated the law. However, if the market value should fall below the assessed value the homeowner may petition the tax collector to lower the assessed value.
Petitioning the value is really a simple process. There is usually a certain time during the year where the appraisal district will hear your petition and render a decision.
Homeowners like when their homes' "market values" steadily increase every year, because it helps to raise their net worth. These same homeowners dread when the "assessed values" of their homes also increase, which most likely translates into higher property taxes.