Concerning the problem of information aggregation in the intuitionistic fuzzy environment, a new Generalized Intuitionistic Fuzzy Geometric Bonferroni Mean (GIFGBM) operator was proposed on the basis of the Archimedean T-norm and S-norm. The proposed operator considered the importance of each attribute and could capture the interrelationships among attributes. Firstly, based on the intuitionistic fuzzy operational laws with Archimedean T-norm and S-norm, the GIFGBM was investigated. Then, its desirable properties were studied, including idempotency, monotonicity, boundedness and permutation invariance. Some special cases of the GIFGBM were further discussed in detail. Finally, an approach to intuitionistic fuzzy multi-attribute decision making was developed with the proposed aggregation operator, and the presented method was applied to research on the development of the regional economy. The experimental results show that the proposed decision making method is practical and effective, and the decision makers can make decision by their attitude.