Before you try to sell your car, you should ask an expert to assess its value — once you know what it's worth, it's easier to find a fair price. When you assess a matter, you make a judgment about it.
The verb assess has the general meaning of determining the importance or value of something. It also has a few specialized uses having to do with amounts of money, such as fines, fees, and taxes. It can mean to set the value of property for purposes of taxation, or to charge a person or business a tax or fee. Assess comes from Anglo-French usage, and is ultimately derived from the Latin verb assidēre, "to sit as a judge."