In this paper, we study the structure of the Hardy space H p b associated with a para-accretive function b, which was introduced in Han et al. (J. Geom. Anal. 14:291-318, 2004). The main result is to establish a new atomic decomposition of H p b. As applications, we obtain criterions for the boundedness of operators on H p b and a characterization of the dual space of H p b. The classification of H p b and Calderón-Zygmund decomposition are also discussed.