There exists a unitary normal vector field on every local chart of a manifold

If regular surface S is connected, and if you have two unitary normal vector fields, then they are either the same or one is the negative of the other

Surfaces for which one can find a unitary normal vector field (remember that a vector field is defined to be differentiable) globally are called Orientable surfaces, otherwise, they are called Nonorientable surfaces. See video