What do the u and v in this slide mean? Are they the coordinates in the texture space?

I don't think they need to be the coordinates in texture space, they are just a general way of parametrizing a curve.

I'm a bit confused. Explicit is supposed to be directly defined but it seems like they are still defining an equation that defines which points are on the object which seems alot like an implicit representation.

Same, I'm still really confused by the difference between explicit and implicit surfaces. Could somebody shed some light on how to tell the difference?

My understanding is that implicit forms do not have such parameterizations, thus making it difficult to determine points on the curve (e.g. guess a point). Explicit forms make this task much easier (choose any u, v in the parameterized range and you'll have a point).

From what i understand explicit is just a finite collection of points/data whereas implicit describes a relationship