  We introduce appropriate computable moduli of smoothness to characterize the
rate of best approximation by multivariate polynomials on a connected and
compact $C^2$-domain $\Omega\subset \mathbb{R}^d$. This new modulus of
smoothness is defined via finite differences along the directions of coordinate
axes, and along a number of tangential directions from the boundary. With this
modulus, we prove both the direct Jackson inequality and the corresponding
inverse for the best polynomial approximation in $L_p(\Omega)$. The Jackson
inequality is established for the full range of $0<p\leq \infty$, while its
proof relies on a recently established Whitney type estimates with constants
depending only on certain parameters; and on a highly localized polynomial
partitions of unity on a $C^2$-domain which is of independent interest. The
inverse inequality is established for $1\leq p\leq \infty$, and its proof
relies on a recently proved Bernstein type inequality associated with the
tangential derivatives on the boundary of $\Omega$. Such an inequality also
allows us to establish the inverse theorem for Ivanov's average moduli of
smoothness on general compact $C^2$-domains.
