Punnett squares are a popular method of showing all the combinations of gametes from sexual reproduction. The gametes are placed in the column or row headings (light or dark gray), and the resulting zygotes are found in the body of the table (purple or white cells). The haploid gametes fuse to make new diploid cells.
Shown at left is a cross between two true breeding individuals. Allele A is dominant; all offspring, being heterozygous, show the dominant trait.
At the right is a monohybrid cross between two heterozygous individuals (perhaps they resulted from crossing the two true-breeding parent stocks at left).
Note that Reginald Punnett devised this method of visualizing progeny by using Mendel's concepts of inheritance. It is for this reason that "Punnett squares" always have a capital "P" - concepts, theories, and structures that are named after the inventor or discoverer are always capitalized (e.g. Krebs cycle).
Mendel observed a 3:1 ratio of offspring from crosses made from parents that differ in a single trait. Purple flowers crossed with white flowers, for example. He also looked at plant height, seed colour, pod colour, and other features. His experiments all showed that one trait dominated the other in the initial batch of offspring (F1, for "first filial generation"). The recessive trait - the one that was not seen in the F1- showed up again when the F1 plants were themselves crossed. Thus, Mendel speculated that inheritance was particulate - the inheritance factor was unchanged, re-emerging in the subsequent generation when two "weak" information particles came together.
The top-left and image to the direct right are testcrosses: one of the parents is homozygous recessive. This parent is the "tester". The offspring that result give clues about the parent with the dominant trait. At top left, it's homozygous dominant. To the right, it's a het. Note that the 50:50 ratio at right wouldn't tell you which trait is dominant. More information is needed!