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In this paper, we have introduced a new intuitionistic fuzzy (IF) entropy called intuitionistic fuzzy entropy of order-α in the settings of intuitionistic fuzzy set theory. It considers both the uncertainty and hesitancy degree of IF sets. Also, we have shown that the entropy suggested by Vlachos and Sergiadis is the particular case of the proposed entropy which does not satisfies the maximality property. Further we have proved the validity of the proposed intuitionistic fuzzy entropy. Some of the properties of the proposed entropy are also discussed and proved that the maximum and minimum values of the proposed entropy are independent of α. At last, application of the proposed entropy is given in multiple attribute decision making (MADM) problem. For this purpose, we have taken a case study on insurance companies.