Active Brownian motion with intermittent direction reversals is common in bacteria like Myxococcus xanthus and Pseudomonas putida. We show that, for such a motion in two dimensions, the presence of the two timescales set by the rotational diffusion constant D_{R} and the reversal rate γ gives rise to four distinct dynamical regimes: (I) t≪min(γ^{-1},D_{R}^{-1}), (II) γ^{-1}≪t≪D_{R}^{-1}, (III) D_{R}^{-1}≪t≪γ^{-1}, and (IV) t≫max(γ^{-1}, D_{R}^{-1}), showing distinct behaviors. We characterize these behaviors by analytically computing the position distribution and persistence exponents. The position distribution shows a crossover from a strongly nondiffusive and anisotropic behavior at short times to a diffusive isotropic behavior via an intermediate regime, II or III. In regime II, we show that, the position distribution along the direction orthogonal to the initial orientation is a function of the scaled variable z∝x_{⊥}/t with a nontrivial scaling function, f(z)=(2π^{3})^{-1/2}Γ(1/4+iz)Γ(1/4-iz). Furthermore, by computing the exact first-passage time distribution, we show that a persistence exponent α=1 emerges due to the direction reversal in this regime.