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$structures. 4 typo Your main question was answered by Emerton. Regarding other notions of orientability, there's many. A popular one is the obstruction-theoretic approach: 1) A manifold$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$TM_{M^0}$that extends over$M^1$. 2) You can restate that definition to one 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$T_pM$. 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. 3 not on relative merits of some constructions 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$TM_{M^0}$that extends over$M^1$. 2) You can restate that definition to one 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$T_pM$. 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.