Moduli space of curves parametrizes genus \(g\) curve with \(n\) distinct point.
Moduli space of stable curves \(\overline{\mathcal{M}_{g,n}} (X)\) parametrizes genus \(g\) stable curve with \(n\) distinct point.
Moduli space of stable maps \(\overline{\mathcal{M}_{g,n}} (X, \beta)\) parametrizes stable maps to \(X\) with homology image with a genus \(g\) curve with \(n\) distinct points.