You want to prove a non-approximability result of an optimization problem, for instance let us consider a minimisation problem, the same for a maximization problem. So you have this problem and you want to prove that this problem not approximable within a given ratio. To this aim you can choose one NP-complete decision problem P2. What you do is to reduce basically P2 to the approximate solution of P1.
In which way? You define a function f that maps instances of the decision problem into instances of the optimization problem, such that if the original instance is a positive instance then the optimum value of the final instance is a given value c(x), if on the other hand if the original instance is a negative instance, then the optimal value must be at least this value plus the gap. This is the gap that we are creating between YES-instances and NO-instances.
