Sexual reproduction is the basis of eukaryotic genetics. Diploid parents make haploid gametes.  These gametes combine at random to make progeny (children). 

The predictive power of genetics comes from the muliplication of the gametic array from the mother with the gametic array of the father.

Example: Consider the cross of two individuals both with genotype Aa.

Aa x Aa

The mother (written first) has the gametic array of :

1/2A + 1/2a = 1

The father (written second) also has the gametic array of :

1/2A + 1/2a = 1.

The genotypic array expected of their children is the product of the two arrays.

Genotypic Array = (1/2A + 1/2a)* (1/2A + 1/2a)

Genotypic Array = 1/4AA + 1/4Aa + 1/4Aa + 1/4aa

(add similar terms) 

Genotypic Array = 1/4AA + 1/2Aa + 1/4aa

The Punnett Square

The Punnett square method is one of the most widely recognized procedures in genetics.  The gametes of the father  are written across the top and the gametes of the mother are written along the left side.  The gametes are then multiplied and written in the cells of the table.  Similar genotypes are then added.

What is seldom appreciated is that the Punnett Square is merely a graphical version of multiplying gametic arrays. The father's gametic array is written at the top, the mother's gametic array is written along the side.  Each term of the father's array is multiplied by each term of the mothers array. Similar genotypes (terms) are then added.

Punnett Square for the F₂ of a dihybrid cross: