We introduce a new iteration method called Picard-S iteration. We show that the Picard-S iteration method can be used to approximate fixed point of contraction mappings. Also, we show that our new iteration method is equivalent and converges faster than CR iteration method for the aforementioned class of mappings. Furthermore, by providing an example, it is shown that the Picard-S iteration method converges faster than all Picard, Mann, Ishikawa, Noor, SP, CR, S and some other iteration methods in the existing literature when applied to contraction mappings. A data dependence result is proven for fixed point of contraction mappings with help of the new iteration method. Finally, we show that the Picard-S iteration method can be used to solve differential equations with retarded argument.