  In this paper we study the following class of fractional relativistic
Schr\"odinger equations: \begin{equation*} \left\{ \begin{array}{ll}
(-\Delta+m^{2})^{s}u + V(\varepsilon x) u= f(u) &\mbox{ in } \mathbb{R}^{N}, \\
u\in H^{s}(\mathbb{R}^{N}), \quad u>0 &\mbox{ in } \mathbb{R}^{N}, \end{array}
\right. \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (0,
1)$, $m>0$, $N> 2s$, $(-\Delta+m^{2})^{s}$ is the fractional relativistic
Schr\"odinger operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a
continuous potential satisfying a local condition, and $f:\mathbb{R}\rightarrow
\mathbb{R}$ is a continuous subcritical nonlinearity. By using a variant of the
extension method and a penalization technique, we first prove that, for
$\varepsilon>0$ small enough, the above problem admits a weak solution
$u_{\varepsilon}$ which concentrates around a local minimum point of $V$ as
$\varepsilon\rightarrow 0$. We also show that $u_{\varepsilon}$ has an
exponential decay at infinity by constructing a suitable comparison function
and by performing some refined estimates. Secondly, by combining the
generalized Nehari manifold method and Ljusternik-Schnirelman theory, we relate
the number of positive solutions with the topology of the set where the
potential $V$ attains its minimum value.
