  Let $\phi: \mathbb{R}^n\times[0,\fz)\rightarrow[0,\fz)$ be a function such
that $\phi(x,\cdot)$ is an Orlicz function and $\phi(\cdot,t)\in
A^{\mathop\mathrm{loc}}_{\infty}(\mathbb{R}^n)$ (the class of local weights
introduced by V. S. Rychkov). In this paper, the authors introduce a local
Hardy space $h_{\phi}(\mathbb{R}^n)$ of Musielak-Orlicz type by the local grand
maximal function, and a local $\mathop\mathrm{BMO}$-type space
$\mathop\mathrm{bmo}_{\phi}(\mathbb{R}^n)$ which is further proved to be the
dual space of $h_{\phi}(\mathbb{R}^n)$. As an application, the authors prove
that the class of pointwise multipliers for the local
$\mathop\mathrm{BMO}$-type space $\mathop\mathrm{bmo}^{\phi}(\mathbb{R}^n)$,
characterized by E. Nakai and K. Yabuta, is just the dual of
$L^1(\rn)+h_{\Phi_0}(\mathbb{R}^n)$, where $\phi$ is an increasing function on
$(0,\infty)$ satisfying some additional growth conditions and $\Phi_0$ a
Musielak-Orlicz function induced by $\phi$. Characterizations of
$h_{\phi}(\mathbb{R}^n)$, including the atoms, the local vertical and the local
nontangential maximal functions, are presented. Using the atomic
characterization, the authors prove the existence of finite atomic
decompositions achieving the norm in some dense subspaces of
$h_{\phi}(\mathbb{R}^n)$, from which, the authors further deduce some
criterions for the boundedness on $h_{\phi}(\mathbb{R}^n)$ of some sublinear
operators. Finally, the authors show that the local Riesz transforms and some
pseudo-differential operators are bounded on $h_{\phi}(\mathbb{R}^n)$.
