I've been reading a little about algebraic geometry and how there seems to have existed this notion of "generic point" on a variety which wasn't carefully defined at first. But often times, intuitively one would try to prove a statement at a generic point to get a statement about the overall variety.
I suppose what I'm asking is: how can I understand the notion of a generic point in the most concrete sense? I know that $X = Spec(\mathbb{Z})$ has $(0)$ as a generic point because it's Zariski closure is all of $X$, but what does that say geometrically?
Suppose I take $V = V(x^2 + y^2 - 1)$ over $\mathbb{R}$, i.e. $Spec(\mathbb{R}[x,y]/(x^2+y^2-1))$. Then $(0)$ is a generic point in the latter (because only maximal ideals correspond to the actual points of $V$, yes?). So how can I visualize this generic point when all I'm thinking about in my mind is the circle in $\mathbb{R}^2$?
As an addendum, what sort of properties did geometers in the early 1900s observe a variety would possess if they could show its "generic point" had this property? (Just as a reference to see where this idea is even coming from, because it feels very unintuitive to me at the moment.) Is there some property about all the other rational primes we can deduce by proving something about $(0) \in Spec(\mathbb{Z})$?