[AussieKenDoll]

What is the meaning of benzenoid and Why it's not possible for both rings on naphthalene to be benzenoid at same time?

[pgk]

This is an issue of Quantum Chemistry.
But roughly, aromaticity is caused by the hybridization of the double bonds. In the benzene ring, all double bonds equally contribute to the hybridization; in contrast to the fused rings, like naphthalene that the double bonds do not equally contribute to the hybridization, which further reflects on the activation/ deactivation of an already substituted naphthalene group.

[hollytara]

If you limit yourself to Lewis structures and resonance structures this is true.  With 10 total pi electrons in the naphthalene system, if one ring "has" 6 and is benzenoid, the other can only have the 4 that are leftover and is more of a "diene".  That is why your organic chem prof will dock you if you draw naphthalene with a 6 pi electron circle in each ring - that means it has 12 pi electrons when it only has 10.  If you are in sophomore organic chemistry - you can stop reading here. 

There is some truth to this - it is easier to hydrogenate the first ring of naphthalene than the second, and far easier than to hydrogenate benzene.  But it is also misleading - the electrons are not required to localize as the Lewis and resonance structures require - they can go wherever they want. 

If you look up a treatment with quantum mechanics / molecular orbital theory you will see how the orbitals stack up by energy.  The lowest orbital has in phase contributions from all carbons around the perimeter, the other 9 orbitals have various combinations.  The lowest 5 are filled, the highest 5 are empty.

[pgk]

BETTER WRITTEN:
Aromaticity is caused by the hybridization of the double bond π orbitals. In the benzene ring, all double bond π orbitals equally contribute to the hybridization; in contrast to the fused rings, like naphthalene that the double bonds π orbitals do not equally contribute to the hybridization, which further reflects on the activation/ deactivation of an already substituted naphthalene group.

[Enthalpy]

[...] both rings on naphthalene [...] at the same time?

Bizarre assertion. Both rings are equivalent, and this is strictly static.

Possibly the claim results from outdated theories that wanted bonds to be local to atom pairs. These wrong theories need bizarre hypotheses and deductions that experiments invalidate, but they are still taught.