What is the difference between shell, subshell, and orbital?
1 Answer
The Schroedinger equation for an electron bound to a spherically symmetric coulomb potential of a hydrogen like nuclei shows that the wave function of the electron forms standing waves called stationary states. These states are characterized by three quantum numbers.

Principal Quantum Number (
#n# ) :#n=1,2,3, \cdots \infty#
For single electron systems the allowed energy values (energy levels) are determined purely by the principle quantum number. This quantum number can only take integer values starting with#1# with no upper bound. All the electronic states with the same principle quantum number are said to belong to the same shell.
The#n=1# states are labeled Kshell ,#n=2# states are labeled Lshell ,#n=3# states are labeled Mshell ,#n=4# states are labeled Nshell and so on. 
Angular Momentum Quantum Number (
#l# ):#l=0,1,2, ..., n1# .
This quantum number determines the magnitude of the orbital angular momentum of the electron. It can take only integer values starting from 0 but has an upper bound. It can only go up to a number that is one less than the principal quantum number. While the shells are a bigger group of quantum states, this quantum number breaks them into smaller groups of quantum states called subshells. All quantum states with the same orbital angular momentum quantum number are said to belong to the same subshell. The#l=0# states are labeled ssubshell,#l=1# states are labeled psubshell,#l=2# states are labeled dsubshell,#l=3# states are labeled fsubshell and so on. Thus the quantum states belonging to a shell with principal quantum number#n# are divided into#n# subshells.  Magnetic Quantum Number (
#m_l# ):#m_l=l,(l1)\cdots,0,\cdots,+(l1),+l#
This quantum number determines the magnitude of the component of the orbital angular momentum vector of the electron along a reference direction (usually the direction of an applied external magnetic field). This quantum number breaks the subshells into further small groups called orbitals . All the electronic states with the same Magnetic Quantum Number (#m_l# ) belong to the same orbital. A subshell characterized by a angular momentum quantum number#l# has#2l+1# orbitals. Thus the ssubshells have 1 orbital, psubshells have 3 orbitals, dsubshells have 5 orbitals and so on.