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How exactly do vectors, normals and faces relate in surfaces?

I understand that given a curve one can construct some kind of surface by duplicating the curve at evenly spaced values of rotation.

However, I've also read that these are not enough for defining a surface. Additionally one needs normals and faces.

So what are normals and faces used for really?

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Surfaces are spaces which have tangent planes at each point. A normal vector and a point completely determine a plane. Faces of your surface you should think of as patches i.e open sets on your surface. Let $p$ be a point on your surface. Then points on the tangent plane are extremely close to your surface so long as you are in some small neighborhood of $p$ i.e so long as you are in the patch about $p$. The fact this this is not enough to define surface is because a surface is a 2-manifold and one condition for being a manifold is that every point has a neighborhood which is homeomorphic to $\mathbb{R}^2$. Think of rotating a finite curved segment which has self-intersection. Can you see how locally this may not be a manifold (i.e how a cusp may arrive)?

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    What does one need the normal vector for? Why are points not enough? – 2017-02-09
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    Also I understand that given a curve one can create a surface by rotating the curve around an axis. Faces are then formed between adjacent "tiles" of 4 points. But what does one need the normals for then? – 2017-02-09