This article gives a basic definition used in cost-benefit analysis of market transactions.
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The consumer surplus on the purchase of a good is the difference between the value the consumer places on the benefits received from the good, and the amount the consumer spends on the purchase.
Consumer Surplus = (Total value received by consumer) - (Total cost to consumer)
The total value received by a consumer is typically approximated by the highest price the consumer would have been willing to pay for the purchase (i..e, the reservation price -- see consumer's reservation price approximates consumer benefit), and the total cost to the consumer is approximated by the cost of the good, i.e., what the consumer paid to acquire the good.
Interpretation as area
Assuming the law of one price (i.e., there is a single price and all consumers pay the same price), the consumer surplus across all consumers for a good at a given price can be computed as the area of the region between the demand curve and the horizontal line for that price: