In this work, we present a low-rank tensor approach for approximating solutions to the nonlinear Vlasov equation. Our method takes advantage of the tensor-friendly nature of the differential operators in the Vlasov equation to dynamically and adaptively construct a low-rank solution basis through the discretization of the equation and an SVD-type truncation procedure. We utilize finite difference WENO and discontinuous Galerkin spatial discretizations, along with a second-order strong stability preserving multi-step time discretization. To preserve conservation properties, we develop low-rank schemes with local mass, momentum, and energy conservation for the corresponding macroscopic equations. The mass and momentum are conserved using a conservative SVD truncation, while the energy is conserved by replacing the energy component of the kinetic solution with one obtained from a conservative scheme for the macroscopic energy equation. We employ hierarchical Tucker decomposition for high-dimensional problems, and demonstrate the high-order convergence, efficiency, and local conservation properties of our algorithm through a series of linear and nonlinear Vlasov examples.