By the computation of generating functions, we can see that there is an identity between the number of self-conjugate partitions and that of ordinary partitions. In this talk, we give a combinatorial proof of this identity by using the bijection of Wright for the Jacobi triple product identity. We also show the relation between hook lengths of a self-conjugate partition and its corresponding partition via the bijection. At the end, we give new combinatorial interpretations for the Catalan number and the Motzkin number in terms of self-conjugate simultaneous core partitions.