The term constant dollars refers to a metric for valuing the price of something over time, without that metric changing due to inflation or deflation. The term specifically refers to dollars whose present value is linked to a given year. (The principle, of course, can be applied to any currency worldwide, not only American dollars. See inflation adjustment and Real versus nominal value (economics) for more general discussions.)
Constant dollars are used to compare the "real value" of an income or price to put the "nominal value" in perspective. For example, who was making more money, your father who made $5,000 at his first job in 1957, or you when you started at $18,000 in 1986? The answer depends on how much can be purchased by each salary. Is a gallon of gasoline more expensive in 1972 than it is today? It depends on how long a person needs to work to earn the money to buy the gas. Converting the nominal values into constant dollars compares what $5,000 could buy in 1957 to what $18,000 could buy in 1986. It is similar to finding the common denominator when adding or comparing fractions. Is 5/8 bigger than 3/5? Convert them both to 40ths and the first becomes 25/40 and the 2nd 24/40.
The inflation calculator at the Bureau of Labor Statistics shows that $5,000 in 1957 has a value of $19,501.78 in 1986 dollars  or that $18,000 in 1986 has a value of $4,614.96 in 1957 dollars. So dad was making more money, in 1957, even though $18,000 looks larger than $5,000. Any year can be used as a baseline for comparing two years as long as it is consistent. For example, both salaries could be converted into 1970 dollars. Then the $18,000 becomes $6,372.26 in 1970 dollars, and the $5,000 becomes $6,903.91 in 1970 dollars. The relative position stays the same no matter what year is used as a baseline. Also the fraction of (1986 salary)/(1957 salary) remains the same as well when both are in constant dollars.
See also 
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