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Given a partial action $\alpha$ of a groupoid $G$ on a ring $R$, we study theassociated partial skew groupoid ring $R \rtimes_{\alpha} G$, which carries anatural $G$-grading. We show that there is a one-to-one correspondence betweenthe $G$-invariant ideals of $R$ and the graded ideals of the $G$-graded ring $R\rtimes_{\alpha}G.$ We provide sufficient conditions for primeness, andnecessary and sufficient conditions for simplicity of $R \rtimes_{\alpha}G.$ Weshow that every ideal of $R \rtimes_{\alpha}G$ is graded if, and only if,$\alpha$ has the residual intersection property. Furthermore, if $\alpha$ isinduced by a topological partial action $\theta$, then we prove that minimalityof $\theta$ 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 spaceis equivalent to $\alpha$ having the residual intersection property. As anapplication, we characterize condition (K) for an ultragraph in terms oftopological properties of the associated partial action and in terms ofalgebraic properties of the associated ultragraph algebra.

The ideal structure of partial skew groupoid rings with applications to topological dynamics and ultragraph algebras