  Given a partial action $\alpha$ of a groupoid $G$ on a ring $R$, we study the
associated partial skew groupoid ring $R \rtimes_{\alpha} G$, which carries a
natural $G$-grading. We show that there is a one-to-one correspondence between
the $G$-invariant ideals of $R$ and the graded ideals of the $G$-graded ring $R
\rtimes_{\alpha}G.$ We provide sufficient conditions for primeness, and
necessary and sufficient conditions for simplicity of $R \rtimes_{\alpha}G.$ We
show that every ideal of $R \rtimes_{\alpha}G$ is graded if, and only if,
$\alpha$ has the residual intersection property. Furthermore, if $\alpha$ is
induced by a topological partial action $\theta$, then we prove that minimality
of $\theta$ is equivalent to $G$-simplicity of $R$, topological transitivity of
$\theta$ is equivalent to $G$-primeness of $R$, and topological freeness of
$\theta$ on every closed invariant subset of the underlying topological space
is equivalent to $\alpha$ having the residual intersection property. As an
application, we characterize condition (K) for an ultragraph in terms of
topological properties of the associated partial action and in terms of
algebraic properties of the associated ultragraph algebra.
