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Summary of introduction to the complex analytic theory of modular curves and forms:

• Equivalence of the categories of elliptic curves over C and lattices in C.
• From lattices+bases to the action of SL(2,Z) on the upper half plane.
• A fundamental domain for this action.
• The groups Gamma(N), Gamma1(N), Gamma0(N) (and congruence subgroups).
• The (moduli) spaces Y(N), Y1(N), Y0(N) and their interpretations and fundamental domains.
• The natural maps between the moduli spaces.

Literature: any book on modular forms.