Your main question was answered by Emerton. Regarding other notions of orientability, there's many. A popular one is the obstruction-theoretic approach:
1) A manifold $M$ is orientable if the tangent bundle $TM$ admits a trivialization when restricted to a $1$-skeleton of a CW-decomposition of $M$. An orientation of $M$ is taken to be a (homotopy class of) trivialization of $TM_{M^0}$ TM_{|M^0}$ that extends over $M^1$.
2) [Corrected to take into account Chris's comment] You can restate definition 1 in a way that avoids skeleta. A popular one is to define the associated orthogonal (principal) bundle to $TM$, lets call it $O(TM)$. This is the bundle over $M$ whose fibers over points $p \in M$ is the general linear group of isomorphisms between $\mathbb R^m$ and $T_pM$. Then $M$ is orientable if every loop $S^1 \to M$ lifts to a loop $S^1 \to O(TM)$.
3) There's a small variant on these ideas called the "orientation cover", this is a 2-sheeted covering space of $M$, and it is connected if and only if $M$ is non-orientable. This has the additional assumption that $M$ is connected.
4) Another variant on this comes from bundle classifying-space machinery. Every vector bundle has a classifying map $M \to B(GL_m)$, and $GL_m$ has a subgroup of positive-determinant matrices, call it $GL^+_m$. $M$ is orientable if and only if the classifying map $M \to BGL_m$ lifts to a map $M \to BGL^+_m$, and an orientation is a homotopy-class of such lifts (flexible enough to allow homotopy of the original classifying map).
Anyhow, those are a few. There's of course more since all these ideas admit perturbations in various directions. For example, another small variant would be that the 1st Stiefel-Whitney class is trivial. One advantage to approaches (1), (2), (4) is that any of them are natural lead-in to other notions of orientation, like $spin$ or $spin^c$ structures.
