Understanding orientability of surfaces I'm trying to understand the intuitive meaning behind the definition of orientability of 2-dimensional surfaces. I understand the formalities, but I just can't seem to get what exactly it means for a surface to be orientable.  Specifically,  I'd like an intuition that was intrinsic: if I were a little ant living inside a surface S, how would my experiences differ if S was orientable and if it wasn't orientable? 
 A: If you're an ant living on the outer surface of a two-dimensional sphere, can you find a way to walk over its inner surface? Since a sphere has no holes, then the answer is no. If you're outside of the sphere, you're there for life. In terms of vectors, picture this: If, as an ant, you're placing vertical flags over its surface (picture flags as unitary vectors) whenever you wanted, all flags placed this way would be pointing "upwards". Then you may call that direction "up".
And since that direction is "up", there must be down. If, preserving that "upward" direction you already have, suppose you're now an ant living INSIDE the sphere, all flags you can place would be pointing "down". There is no path you can travel on the surface of the sphere that would make you lose that reference: if you're inside, the vertical direction (from the ground up) is "down". If you're outside, the vertical direction is "up". That means you can ORIENT yourself, and that means the sphere is orientable. The notions of "up" and "down" are preserved no matter where you are over the sphere.
Now, you've probably heard that the Moebius strip is not orientable. If you're the ant walking over it and placing flags, suppose that you're always walking in the middle path between the two borders. Eventually you'll be able to place two flags on the same point of the strip, but now one is pointing up and the other is pointing down. Therefore, you can't orient yourself as you did over the sphere, and as such, the Moebius strip is not orientable.
I hope this helped.
A: Note that nonorientability of a surface $S$ is a global property of $S$. Unless you are in mental possession of all of $S$ you cannot intuitively grasp its nonorientability. If a Moebius band is so long that an ant could never make it around $M$ in its lifetime  this ant would never detect mirror-reflected copies of its colleagues, and similarly for an ant on the projective plane $S^2/\pm1$.
Of course orientability can be described in many different ways at various levels of sophistication. One is the following: A surface $S$ is orientable if you can set up an atlas of local coordinates on $S$ such that all  transformations between different charts have positive Jacobian. But it is not easy to prove that, e.g., a Moebius band defined as infinite strip modulo a glide-reflection does not possess such an atlas.
A: Orientation is just your ability to determine position. For surfaces, if you're an ant then locally you are on a plane in which you establish which way is north and which way is east (south, west are determined once this decision is made). Then say you travel along some path and all of a sudden everything switches (i.e up is down and left is right). Therefore if you consider your starting position and ending position (where directions change) you are in disagreement i.e there is no way to determine position. This is an example in which the surface is non-orientable. This is why in the definition we take frames (compass) at a point $p$ and require that the frames vary smooth (in particular, continuously)along the surface. 
Observe this is not the case is our example since you can stand arbitrarily close to where the orientation changes (say this point is $p$) and as soon as you move some $\epsilon$ distance, everything flips (i.e this is not a smooth variance). I hope this makes more sense now, I tried to throw in the general definition to make it clear as to what situation it is trying to capture. This is also completely intrinsic like you wanted. 
A: A surface is orientable iff it does not contain a Möbius strip as a sub-surface.
So it is easier to define not orientable. A surface is not orientable iff it contains a Möbius strip.
Technically a Mobius strip is a regular (tubular) neigborhood of an embedded loop which reverse orientation.
