As for somewhat informal (and mostly brief) introductions to moduli spaces, see

• Introduction to moduli spaces by Han-Bom Moon

• The Moduli Space of Curves and Its Tautological Ring by Ravi Vakil in the Notices of the AMS

• An introduction to the moduli spaces of curves by Maarten Hoeve

• Rational Points on Moduli Spaces of Curves by Dave Jensen

• A minicourse on moduli of curves by E. Looijenga

See also the BAMS paper Perturbations, deformations, and variations (and ``near-misses") in geometry, physics, and number theory by Barry Mazur, in particular, Sections 4 and 5.

As for somewhat informal (and mostly brief) introductions to moduli spaces, see

• Introduction to moduli spaces by Han-Bom Moon

• An introduction to the moduli spaces of curves by Maarten Hoeve (very brief)

• Rational Points on Moduli Spaces of Curves by Dave Jensen (also very brief)

• A minicourse on moduli of curves by E. Looijenga

See also the BAMS paper Perturbations, deformations, and variations (and ``near-misses") in geometry, physics, and number theory by Barry Mazur, in particular, Sections 4 and 5.

As for somewhat informal introductions to moduli spaces, see

• Introduction to moduli spaces by Han-Bom Moon

• An introduction to the moduli spaces of curves by Maarten Hoeve (very brief)

• Rational Points on Moduli Spaces of Curves by Dave Jensen (also very brief)

See also the BAMS paper Perturbations, deformations, and variations (and ``near-misses") in geometry, physics, and number theory by Barry Mazur, in particular, Sections 4 and 5.