The two simplest examples I know are Abelian varieties, which David writes about, and K3 surfaces. Rough sketch for K3s: a 20 dimensional complex moduli space, but the algebraic tangent space to any moduli point is 19 dimensional.

The two simplest examples I know are Abelian varieties (the only algebraic complex Tori, dimension g(g-1)/2 inside the space of complex Tori which is dimension g^2 - David writes about them in his post), and K3 surfaces. Rough sketch for K3s: a 20 dimensional complex moduli space, but the algebraic tangent space to any moduli point is 19 dimensional.