Definition: Let $p: M \rightarrow B$ be a $G$-equivariant fibration of $G$-manifolds. Then the map p is said to have $G$-oriented fibers if the fibers of p are oriented with an orientation varying continuously and if the $G$-action on $M$ preserves the orientation of all the fibers.
What does it mean that the fibers are oriented with an orientation varying continuously ? I'm not sure what is the definition of continuity of an orientation ?
The intuitive idea is that nearby fibers should have "the same" orientation.
More concretely, given a local trivialization $\varphi:\pi^{-1}(U)\to U\times F$, a fiber orientation will induce an orientation $\mathcal{O}_x$ of the typical fiber $F$ for each point $x\in U$. The fiber bundle is continuously oriented on $U$ if $\mathcal{O}_x$ is independent of $x$ on $U$.
In the smooth case, we can equivalently say that a bundle has continuously orientable fibers if there is a smooth differential form $\omega$ such that $\omega|_{\pi^{-1}(p)}$ is an orientation form on each fiber.
As an example, the Möbius strip has orientable fibers, but we cannot choose a continuous fiber orientation.