Utilizing the free Polyakov action of open p-brane, we obtain the solution of satisfying the boundary conditions to Euler-Lagrange equation of the open p-brane, the first and second quantization theories of the p-brane are given. Further, we obtain a series of new multiple commutative relations between the different normal modes of the p-brane, obtain the new lowering and raising multiple operators, and give a series of the new fundamental multiple commutative relations of the lowering and raising multiple operators in the state space. And then we mainly take p = 3 case for example to investigate the spectrum of the open 3-brane at different levels. Interestingly, we find three types of tachyon states including scalar states, vector states and 2-rank tensor states, respectively. Besides, the graviton fields, Kalb-Ramond fields, dilaton and photon states, 3-rank tensor states as well, appear at the same level in the open 3-brane model. For the spectrum of the p-brane (p > 3), one can do the whole analogous research on p = 3 except more complex.