An oriented atlas is a smooth atlas whose smooth transition maps have positive Jacobian determinants on every overlap. The coordinate bases then give compatible choices of orientation of a vector space on all tangent spaces. A nowhere-vanishing top-degree differential form gives such an atlas: restrict each chart until its coefficient has constant sign, reverse one coordinate if necessary to make that sign positive, and compare the positive coefficients on overlaps. Conversely, a partition of unity glues positive local coordinate top forms to a global volume form.
New to topics? Read the docs here!