In electronic spectroscopy of molecules, why some quantum numbers are considered to be 'good quantum numbers'? For example, $n$ and $l$ are said to be not good quantum numbers while $j$ is considered to be a 'good quantum number'. What is the logic/idea behind these concepts?
Good quantum number are associated with operators that commute with the Hamiltonian. They correspond to conseved quantities.
Overall angular momentum is conserved, but the portions of it due to orbital motion and due to spins are not themselves conserved. n is 'bad' in that there's no conserved physical quantity related to radial motion.
A decently good explanation is at http://mxp.physics.umn.edu/s09/projects/S09_OpticalPumping/atomicstructure.htm (A snapshot of the page can be found here)