is an option for DiscretizeRegion and related functions that specifies a function to indicate whether mesh cells should be refined or not.
With MeshRefinementFunction->f, the function f[vlist,m] is applied to each simplex created, where vlist is a list of the vertices and m is the measure. If f[vlist,m] returns True, the simplex will be refined.
open allclose all
Discretize a Disk so that the triangles near the center are smaller:
Discretize a Rectangle so that the triangles in the first quadrant are smaller: