Simply put, Huckel's rule for aromaticity states that a monocyclic system will be aromatic if there are 4n + 2 delocalised electrons, (n an integer) contained within it. e.g.: 2, 6, 10, 14 etc.

Huckel arrived at this rule by performing molecular orbital calculations on cyclic systems containing x carbon atoms, and with each carbon atom supplying one pi electron.  He connected high aromatic stability with the presence of (4n + 2) electrons. 

It is important to note that the number of carbon atoms in the cyclic system, according to this rule, is irrelevant to the aromaticity or otherwise of the molecule; it is purely based on the number of delocalised electrons.

Not mentioned in the rule, but implicitly part of aromaticity is the requirement that the ring is (almost or exactly) planar.  This is simply because the delocalisation only works when there is significant overlap of the p orbitals (as mentioned earlier), and any significant deviations from planarity will diminish the p orbital overlap.

Carbon atoms	3	4	5	6	7	8
Structure
pi electrons	4n + 3	4n	4n + 1	4n + 2	4n + 3	4n

For example, let us consider cyclobutadiene briefly: it has 4n electrons in the pi system, as we have seen.  These occupy energy levels as shown below:

Cyclobutadiene	Benzene

Cyclobutadiene has two bonding electrons, but it also has two unpaired non-bonding electrons.  These electrons do not contribute to the stability of the molecule, and because they are in high energy orbitals, they are available for reactions.

Benzene, on the other hand, has all 6 pi electrons in bonding orbitals, and through this achieves extra stability.