Given an elliptic curve $E/\mathbb{Q}$ of conductor $N$, parameterization $\psi : X_0(N) \rightarrow E$, and a point $P \in E$, take the fiber $\psi^{-1}(P)$. Its points, being on $X_0(N)$, correspond to equivalence classes of pairs $(E_x, C_x)$.

Is there a geometric meaning for these pairs in relation to the point $P$? Something about these elliptic curves that has something to do with $P$? (Other than the j-invariant solving some polynomial equation with coefficients depending only on $P$ (and $E$ of course))

Anything special about the $C_x$'s in relation to $P$?

Modular parameterization is fascinating, but I just don't understand where it comes from. $X_0$ is for a whole bunch of elliptic curves. $E$ is a single specific one.What's the connection between points on the modular curve, to a single specific point of $E$?