Manifolds, Spheres, Parametrizations...

I took a course on manifolds... and I'm trying to put this new information together with what I learned about parametrizations in vector calculus, and I'm having a bit of trouble.

In undergrad vector calculus we talked about parametrizing a sphere, and as far as I could tell, there didn't ever seem to be a parametrization that was smooth for all points (eg. might not be smooth at +/- pi/2).

I also know that if I want to show that the sphere is a manifold, I need at least 2 coordinates charts.

Is there a connection here? I guess I'm trying to put together was it means for a manifold to be smooth (smooth transition functions) versus what it means for a parametrization to be smooth (normal vector is always nonzero).