First of all, the things that you actually integrate are densities, which are the differential geometric counterparts of measures. No orientation is needed. On a smooth manifold $M$ with separable topology there is an intrinsic concept of negligible set. A density is then a signed measure $\mu$ such that $|\mu|(B)=0$ for any negligible Borel subset.
A degree $n$ form on an $n$-dimensional manifold is almost a density, but not quite. We need an orientation to associate to the top degree form a density. This is what you ultimately integrate when you integrate a form. For more details see Section 3.4.1 of these notes.