"Moduli" are parameters that algebraic varieties depend on - continuous invariants if you like, as opposed to discrete invariants. Or the same for complex structures on a given topological manifold, if you like to think that way. A modular curve is the simplest actual case of the phenomenon, the case of a single variable. This was first noticed when the varieties were elliptic curves. There was an elaborate working-out of the theory throughout the nineteenth century: the usual language was of modular equations (in two variables), which after Riemann were probably read by geometers typically as defining Riemann surfaces. There is a lot of organisation for all the possible modular curves coming from classical elliptic function theory.
And in the twentieth century a number of successive points of view were found explaining all this theory again. A sensible place to start is probably Hecke's theory; partly that's a matter of taste, some people might prefer the work of Hurwitz which might wrongly be considered subsumed by now. You don't need the generality of all Fuchsian groups, or all automorphic forms, or all Galois representations, or all L-functions. Modular curve theory can indicate how all of these fit in, as classical examples.