  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 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 (i) Whitney type estimates with constants depending only on
certain parameters; and (ii) highly localized polynomial partitions of unity on
a $C^2$-domain. Both (i) and (ii) are of independent interest. In particular,
our Whitney type estimate (i) is established for directional moduli of
smoothness rather than the ordinary moduli of smoothness, and is applicable to
functions on a very wide class of domains (not necessarily convex). It
generalizes an earlier result of Dekel and Leviatan on Whitney type estimates
on convex domains. The inverse inequality is established for $1\leq p\leq
\infty$, and its proof relies on a new Bernstein type inequality associated
with the tangential derivatives on the boundary of $\Omega$. Such an inequality
also allows us to establish the Marcinkiewicz-Zygmund type inequalities,
positive cubature formula, as well as the inverse theorem for Ivanov's average
moduli of smoothness on general compact $C^2$-domains.
