  Let $O_{m+1}$ denote the Odd graph on a set of cardinality $2m+1$, where $m$
is a positive integer. Denote by $X$ its vertex set and by $T:=T(x_0)$ its
Terwilliger algebra with respect to any fixed vertex $x_0\in X$. In this paper,
we first prove that $T$ coincides with the centralizer algebra of the
stabilizer of $x_0$ in the automorphism group of $O_{m+1}$ by considering the
action of this automorphism group on $X\times X\times X$. Then we give the
decomposition of $T$ for $m\geq 3$ by using all the homogeneous components of
$V:=\mathbb{C}^X$, each of which is a nonzero subspace of $V$ spanned by the
irreducible $T$-modules that are isomorphic. Finally, we display an orthogonal
basis for every homogeneous component of $V$.
