TITLE: How should I think about what it means for a manifold to be orientable?
QUESTION [22 upvotes]: Let M be a smooth manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ such that for all $\alpha, \beta$, $\textrm{det}(J(\phi_{\alpha} \circ \phi_{\beta}^{-1})) > 0$ (where defined). I'm struggling to understand the reason this definition is made. My question is: What is the intuitive reason for this definition of an orientable manifold? 
Thank you!

REPLY [5 votes]: I think it is very hard to get a better answer than any of the above. But there is a perspective that has not been mentioned, at least not explicitly. First, I would like to say that I happen to agree with Marianos answer.
The body of your question is much more explicit than your title, so I will address the question in your title since it seems others have addressed the one in the body of your question. I like to think of orientations, or in fact most structures on a manifold, in terms of the tangent bundle on the manifold in question. The tangent bundle on a compact n dimensional manifold $M$ is classified (up to isomorphism) by a homotopy class of map $M \to BO(n)$. If we can lift this map over the map $BSO(n) \to BO(n)$ then the bundle is orientable, and hence so is the manifold. Here, the notion of orientability, our structure is really a fact about the tangent bundle. We can talk about almost complex structures or spin structures in a similar way. The map $BSO(n) \to BO(n)$ is induced by the inclusion $SO(n) \to O(n)$.
Another way is to look at $\Lambda^n TM$, the top exterior power of the tangent bundle of M. Then an orientation is a nowhere zero continuous section of this bundle. Similarly, you can think of an orientation as a choice of generator of the top dimensional integral homology or cohomology. These are all probably less helpful than the above answers, but these are the ways I like to think about what an orientation is.
please let me know if you want me to expand on some of these.