Not sure if it's essentially* the same as the already given answer(s), but a useful definition of orbifolds and morphisms thereof (that works also in other categories) is via [stacks][1].

I think in this framework a (topological) orbifold would be a stack on the category of (not necessarily compact) topological manifolds that is locally isomorphic to a quotient stack of the form $[\mathbb{R}^n/G]$ with $G$ a finite group (perhaps embeddable in $O(n)$ for the same $n$, if you want). Note that the notion of sheaf and the analogous notion of "vector bundle" is very naturally available in this framework. There is also the analogous theory of differentiable stacks and in particular differentiable orbifolds, where you can talk about tangent "bundles" and metrics.

${}^*$ For a groupoid internal to a category of "spaces" is some kind of "presentation" for a stack on the same category, and Morita equivalence corresponds to the right notion of "isomorphism" (equivalence) of stacks.

  [1]: http://en.wikipedia.org/wiki/Stack_%28mathematics%29