"Aromaticity" is a term coined by Hofmann in 1855 to describe a group of chemicals. This term relating a chemical to their aroma is extremely misleading as there is not a general relationship between the chemical property of aromaticity and the olfactory properties of aromatic compounds. In reality, "aromaticity" describes a conjugated ring (system) of unsaturated bonds, lone pairs, or empty orbitals; or rather the delocalization of electrons about a cyclic structure due to resonance. These compounds are not prone to electrophilic addition like alkenes are because of the extreme stablility that lies within the delocalization of electrons, but rather undergo electrophilic and nucleophilic substitution.
Hückel aromaticity originates from German physicist Erich Hückel. In 1931, he proposed his "4n+2" rule, which he based off the Molecular Orbital Theory since a compound with all its bonding molecular orbitals filled with electrons are stable. Using this theory in aromatic compounds, 2 electrons fill the lowest energy molecular orbital while 4 electrons fill each subsequent energy level (n=# of susequent levels). This method leaves all bonding orbitals filled with no anti-bonding orbitals occupied.
Möbius aromaticity on the other hand, views orbitals in the form of a Möbius strip, yielding its name. In these systems, the pattern of orbital energies is partially inverted from that of Hückel aromaticity, resulting in the "4n" rule instead.