This paper studies the local controllability for a class of multidimensional quasilinear parabolic equations with homogenous Dirichlet boundary conditions and an arbitrarily located internal controller. Unlike the known result in one spacial dimension, controllability is analyzed in the frame of classical solutions. The key of our approach is to seek a control function in the Holder space for given data with certain regularity. In order to give the cost estimate for the control function, we establish an observability inequality for linear parabolic equations with an explicit estimate on the observability constant in terms of the C-1 coefficients of principal parts. The later is based on a new global Carleman estimate for general linear parabolic equations, which has an independent interest.