We present a solver for the Poisson-Boltzmann equation and demonstrate its applicability for biomolecular electrostatics computation. The solver uses a level set framework to represent sharp, complex interfaces in a simple and robust manner. It also uses non-graded, adaptive octree grids which, in comparison to uniform grids, drastically decrease memory usage and runtime without sacrificing accuracy. The basic solver was introduced in earlier works [16, 27], and here is extended to address biomolecular systems. First, a novel approach of calculating the solvent excluded and the solvent accessible surfaces is explained; this allows to accurately represent the location of the molecule's surface. Next, a hybrid finite difference/finite volume approach is presented for discretizing the nonlinear Poisson-Boltzmann equation and enforcing the jump boundary conditions at the interface. Since the interface is implicitly represented by a level set function, imposing the jump boundary conditions is straightforward and efficient.