whats the definition of a conformal mesh?

A conformal mesh is a mesh where the geometry and order lines up between elements (eg. no hanging nodes and constant polynomial order). h-nonconforming means there are hanging nodes and p-nonconforming means that polynomial order doesn't match. A conformal discretization is one where the global function space has the expected regularity (typically H^1). Discontinuous Galerkin uses a nonconforming discretization since the global function space is not in H^1 (although it is H^1 on elements). You can use a conformal discretization on a nonconforming mesh by using constraints and eliminating constrained degrees of freedom.

I think you may need to supply some context. A common meaning is constant scale factors.