A surface becomes a Riemann surface when it is endowed with an additional geometric structure.
English is not my native language, and I cant understand what does endowed mean here.
Endowed can also be replaced with equipped there. It means the surface is given some extra structure. That is, some extra structure is defined on it.
Mathematical structure is translated oddly into English. The words used don't invoke an object, but sound more like a cooking recipe: e.g. to define a group, you start with a set and then you mix in additional ingredients (a choice of unit, an inversion function, and a product function), and the end result is a group.
"Endowed" is just one of the many words that people use to express the English idea of starting with something and adding something more.
I'd also suggest that this is one of those terms where mathematical terminology and popular usage are somewhat opposite.
In a populist sense it can mean 'is restricted to' (rather than having open ended possibilities). Of course mathematically, there can be no possibilities without some extra axiom or endowment.