The most well-known aromatic molecule is benzene, so we shall start with a brief discussion of this molecule.  It must be made clear at this point that "aromatic" here refers to a molecule with certain electronic properties, it does not mean a molecule that smells (although that is in fact where the name first came from)!

The structure of benzene was a problem for the early chemists.  It was known that benzene had the formula C₆H₆, and that it was relatively unreactive.  It was also known that benzene did not perform the familiar addition type reactions of alkenes, but performed substitution with e.g. Br₂ in the presence of FeBr₃.

The addition product, which is not formed, would be C₆H₄Br₂.

Using this and other evidence,  Kekulé proposed the following famous structure of benzene:

Although his proposed structure did account for the fact that there is only one isomer of C₆H₅Br, it did not explain why benzene was so unreactive, and why it performed substitution reactions as opposed to addition.

However, using the concept of resonance gives us a clue as to the answer to these questions.  If we consider the following resonance forms:

We can see that they are in fact, entirely equivalent, and that a better representation might be:

Here we can see the 6 sp2 carbon atoms.  Only their p orbitals are shown, and we can see that they overlap with both their neighbours. (Compare Kekulé's structure,which effectively considers each p orbital asoverlapping with one other.  Hence, all the pi electrons are delocalised in rings above and below the plane of the molecule.

This resonance proposal also helps explain another problem with Kekulé's structure: from what we know about hybridisation, we would expect that the double bonds in his structure of benzene would be significantly shorter than the single bonds.  Normal double bonds are of the order of 1.3Å, and single bonds are of the order of 1.5Å.  However, upon examination, it is found that all the bonds in benzene are the same length, at roughly 1.4Å.  This makes sense under our proposed resonance scheme, but not under Kekulé's structure.