Properly discontinuous action This is exercise 9 of chapter 0 from Do Carmo's book in Riemannian Geometry:
Let $G\times M \rightarrow M$ a properly discontinuous action from a Group $G$ on a smooth manifold $M$. Prove that $\frac{M}{G}$ is orientable if and only if there is a orientation of $M$ who is preserved by all diffeomorphism of $G$

I don't know exactly where your confusion lies, so I'll give a quick (and possibly unhelpful) hint and elaborate if it isn't enough. 
The action of $G$ on $M$ induces a smooth manifold structure on $M/G$ - I'm assuming you understand how this works. If $M/G$ is orientable, then choose any oriented atlas $\{U_\alpha\}$ on $M/G$ and consider the cover $\{\pi^{-1}(U_\alpha)\}$ of $M$ (where $\pi \colon M \to M/G$ is the quotient map). Can you create an oriented atlas on $M$ using this cover?
Conversely, if $M$ has an oriented atlas $\{V_\alpha\}$, you can consider the cover $\{\pi(V_\alpha)\}$ of $M/G$, but in general this won't give you an atlas of $M/G$ unless the atlas $\{V_\alpha\}$ somehow comes from the action of $G$ on $M$.
Where exactly are you stuck?
