In terms of molecular bonding and organic chemistry the term aromatic refers to the delocalisation of the electrons in the additional bonds of covalent double bonded carbon compounds such as benzene.

If there are two or more double bonds in a molecule and they correctly aligned (basically they are separated by one single bond and the double bonds are co-planar) then the bonding orbitals can overlap and the electrons can be delocalised over all the double bonds.

The bonds of the aromatic system are not single or double bonds but seem from length etc. to be a hybrid. This is particularly important as the aromatic double/single bonds are more stable than normal. This is because for them to react not only would the bond have to be broken but the extended aromatic system would have to be disrupted. This takes more energy. The best example of an aromatic compound is the ring compound benzene.