The English Wikipedia introduces orientability with this:
In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point. A choice of a normal vector allows one to use the right-hand rule to define a “clockwise” direction of loops in the surface, as needed by Stokes' theorem for instance. More generally, orientability of an abstract surface, or manifold, measures whether one can consistently choose a “clockwise” orientation for all loops in the manifold.
I understood most, but not all of this. To non-technical people, I’ll bet it reads like French. Unless you can read French. Omelette du fromage.
I say this as a regular Wikimedia Foundation donor and Wikipedia writer myself: Wikipedia’s mathematics articles are obtuse to the point of uselessness for the majority of the site’s potential readers. Aka, people doing web searches for something they’re trying to learn, because they’ve been told to RTFM instead of asking questions.
By all means have detail later in the article, but I’ve almost never been impressed by any mathematics article page lead. Compare this to most of the chemistry articles for example, and the difference is night and day.
And before the actually crowd attacks, this is not what the Simple English Wikipedia is for.