The Atanassov’s intuitionistic fuzzy set (IFS) has become a popular topic of investigation in the fuzzy set community. However, there is less investigation on the representation of level sets and extension principles for Atanassov’s IFSs as well as algebraic operations. In this paper, firstly the representation theorem of Atanassov’s IFSs is proposed by using the concept of level sets. Then, the extension principles of Atanassov’s IFSs are developed based on the representation theorem. Finally, the addition, subtraction, multiplication and division operations over Atanassov’s IFSs are defined based on the extension principle. The representation theorem and extension principles as well as algebraic operations form an important part of Atanassov’s IFS theory.