We examine the integral cohomology rings of certain families of $2n$-dimensional orbifolds $X$ that are equipped with a well-behaved action of the $n$-dimensional real torus. These orbifolds arise from two distinct but closely related combinatorial sources, namely from characteristic pairs $(Q,\lambda)$, where $Q$ is a simple convex $n$-polytope and $\lambda$ a labelling of its facets, and from $n$-dimensional fans $\Sigma$. In the literature, they are referred as toric orbifolds and singular toric varieties respectively. The first main result provides combinatorial conditions on a characteristic pair $(Q, \lambda)$ which ensure that the integral cohomology groups of a toric orbifold are concentrated in even degrees and torsion free. The second main result assumes these condition to be true, and expresses the cohomology ring of a toric orbifold as a quotient of a polynomial algebra that satisfies a certain condition called integrality condition arising from the underlying combinatorial data. Finally, we extend the idea of result about toric orbifolds to the category of torus orbifolds. We introduce the \emph{orbifold torus graph} which is an extension of the torus graph and the GKM graph. By the aids of Chang--Skjelbred sequence, we compute the equivariant cohomology ring with integer coefficients of a torus orbifold from the torus orbifold graph.