This work proposes and analyzes a new local projection stabilized (LPS for short) finite element method for the nonlinear incompressible Navier-Stokes equations. Stokes problems defined element-wisely drive the construction of the stabilized terms which make the present method stable for P-1 x P-2, for continuous pressure and P-1 x P-0 for discontinuous pressure, in two- and three-dimensions. Existence and uniqueness of a discrete solution and a nonsingular branch of solutions are proved under standard assumptions. Also, we establish that the LPS method achieves optimal error estimates in the natural norms. Numerics assess the theoretical results and validate the LPS method in the three-dimensional case.