An aromatic (or aryl) compound contains a set of covalently bound atoms with specific characteristics:
# A delocalized conjugated π system, most commonly an arrangement of alternating single and double bonds
# Coplanar structure, with all the contributing atoms in the same plane
# Contributing atoms arranged in one or more rings
# A number of π delocalized electrons that is even, but not a multiple of 4. That  is, 4n + 2 number of π electrons, where n=0, 1, 2, 3, and so on.  This is known as Hückel's Rule.
Whereas benzene is aromatic (6 electrons, from 3 double bonds), cyclobutadiene is not, since the number of π delocalized electrons is 4, which of course is a multiple of 4.  The cyclobutadienide (2−) ion, however, is aromatic (6 electrons). An atom in an aromatic system can have other electrons that are not part of the system, and are therefore ignored for the 4n + 2 rule.  In furan, the oxygen atom is sp² hybridized. One lone pair is in the π system and the other in the plane of the ring (analogous to C-H bond on the other positions). There are 6 π electrons, so furan is aromatic.
Aromatic molecules typically display enhanced chemical stability, compared to similar non-aromatic molecules. A molecule that can be aromatic will tend to alter its electronic or conformational structure to be in this situation.  This extra stability changes the chemistry of the molecule. Aromatic compounds undergo electrophilic aromatic substitution and nucleophilic aromatic substitution reactions, but not electrophilic addition reactions as happens with carbon-carbon double bonds.
Many of the earliest-known examples of aromatic compounds, such as benzene and toluene, have distinctive pleasant smells. This property led to the term "aromatic" for this class of compounds, and hence the term "aromaticity" for the eventually discovered electronic property.
The circulating π electrons in an aromatic molecule produce ring currents that oppose the applied magnetic field in NMR. The NMR signal of protons in the plane of an aromatic ring are shifted substantially further down-field than those on non-aromatic sp² carbons.  This is an important way of detecting aromaticity.  By the same mechanism, the signals of protons located near the ring axis are shifted up-field. 
Aromatic molecules are able to interact with each other in so-called π-π stacking: The π systems form two parallel rings overlap in a "face-to-face" orientation. Aromatic molecules are also able to interact with each other in an "edge-to-face" orientation: The slight positive charge of the substituents on the ring atoms of one molecule are attracted to the slight negative charge of the aromatic system on another molecule.
Planar monocyclic molecules containing 4n π electrons are called antiaromatic and are, in general, destabilized.  Molecules that could be antiaromatic will tend to alter their electronic or conformational structure to avoid this situation, thereby becoming non-aromatic.  For example, cyclooctatetraene (COT) distorts itself out of planarity, breaking π overlap between adjacent double bonds. Relatively recently, cyclobutadiene was discovered to adopt an asymmetric, rectangular configuration in which single and double bonds indeed alternate; there is no resonance and the single bonds are markedly longer than the double bonds, reducing unfavorable p-orbital overlap. This reduction of symmetry lifts the degeneracy of the two formerly non-bonding molecular orbitals, which by Hund's rule forces the two unpaired electrons into a new, weakly bonding orbital (and also creates a weakly antibonding orbital). Hence, cyclobutadiene is non-aromatic; the strain of the asymmetric configuration outweighs the anti-aromatic destabilization that would afflict the symmetric, square configuration.