An Organic compound that contains at least one cyclic (ring-shaped), planar (flat) Moiety with a ring of resonance bonds (most often a Conjugated system) that exhibits more stability than other geometric or connective arrangements with the same set of atoms. Aromatic moieties are very stable and do not easily break apart and react with other substances.
The most common of these rings is the Benzene ring.
Existence of such a stable resonant cyclic can often be determined using Huckel's rule
Aromatic Compounds and Huckel's Rule
There is a class of resonant rings that are very unstable, known as Antiaromatic compounds. Finally there are resonant rings which are non-aromatic (neither aromatic nor antiaromatic).
Aromatic, Antiaromatic, or Nonaromatic: How To Tell The Difference