"Non-orientable surface" is the thing. The surface of that thing is not orientable. That means you can't draw some normal vector to define the surface. When you traverse through the surface with a normal vector you can end up in the same point with that vector pointing the opposite direction in such a surface. That gives two directions for the same point or in other words you can't define the surface like that with a normal vector. Its the surface and not the shape itself that is non-orientable

firstly understand the context which is smooth manifolds, for simplicity imagine a 2d manifold embedded in 3d space - so a sheet of rubber that can pass through itself but can't kink or do any funny business, just like in that sphere inversion video.

the definition of a manifold is basically that it can be built out of patches (sheets of rubber in our analogy), for instance to make a sphere, we need two sheets of rubber (ignore the actual logistics of the deformation required).

Now say that our sheets of rubber come with a textured and a smooth side, there are two ways to attach the sheets of rubber to make a sphere, one of which produces a sphere which is entirely smooth on the outside. This is what we mean by orientable, we can build it out of patches with a consistent "outside".

Consider the counterexample of a mobius strip, which we construct from a single strip of rubber by attaching one end to the other "backwards" (rough-smooth). Since we have defined it this way, it cannot be orientable. The klein bottle is another example, but somewhat cooler than the mobius strip since its a surface without edges.

There are many other definitions of orientable depending on the context, since manifolds are a lot more general than I have shown you here.

I don't know what orientable manifolds have to do with being responsible.

If you follow the surface with your finger, starting from the outside, you can end up on the inside, without traversing the surface. There is only one surface, no concept of outside or inside, contrary to a good old cube for instance.

Similar to a Moebius strip but with a higher dimension. A MS is easier to understand since you can easily make one to try to run your finger on the surface and end up anywhere.