  Let $G$ be a finite, simple, connected graph. An arithmetical structure on
$G$ is a pair of positive integer vectors $\mathbf{d},\mathbf{r}$ such that
$(\mathrm{diag}(\mathbf{d})-A)\mathbf{r}=0$, where $A$ is the adjacency matrix
of $G$. We investigate the combinatorics of arithmetical structures on path and
cycle graphs, as well as the associated critical groups (the cokernels of the
matrices $(\mathrm{diag}(\mathbf{d})-A)$). For paths, we prove that
arithmetical structures are enumerated by the Catalan numbers, and we obtain
refined enumeration results related to ballot sequences. For cycles, we prove
that arithmetical structures are enumerated by the binomial coefficients
$\binom{2n-1}{n-1}$, and we obtain refined enumeration results related to
multisets. In addition, we determine the critical groups for all arithmetical
structures on paths and cycles.
