The multi-attribute decision-making has the problems that the decision information is Intuitionistic Fuzzy Set (IFS) and the attribute weights are completely unknown. In order to solve the problems, a decision-making method based on Intuitionistic Fuzzy (IF) entropy and co-correlation degree was proposed. Considering the intuitionism and fuzziness of IFS, an improved IF entropy was defined from the axiomatic definition. Furthermore, based on the criterion that the total uncertain information of all the attributes kept minimization, a nonlinear programming model was established by utilizing the proposed IF entropy, and the formula of attribute weights was obtained. From the structure of the correlation coefficients in statistics between the variables, the concept of co-correlation degree of IFS was proposed, and the similar properties with correlation coefficient were also discussed. Moreover, the formula of co-correlation degree weighted between each object and the ideal object was acquired. Finally, a new multi-attribute decision-making approach was presented, which was successfully applied to the example of teachers' election. By calculating the co-correlation degree of each teacher, the best candidate was determined, and the optimal decision was achieved. With the advantages of reasonable operation, reliable calculation result, and easy to implement, the proposed method can be used for a variety of decision problems.