To illustrate how a Punnett square works, let's look at the experiments done by "The Father of Genetics" Gregor Mendel's work with pea plants. 

The pea plants Mendel studied had two alleles for height: a dominant tall allele (T) and a recessive short allele (t). Since pea plants produce using sexual reproduction each pea plant puts one of its two alleles for height into every gamete that it makes. Each allele for height has an equal chance of becoming part of a gamete, so half of the gametes carry one allele, and half of the gametes carry the other allele. 

In a Punnett square, the alleles of the parents are placed on the top and side of the square. So if one plant is homozygous dominant for height (TT), and the other plant is homozygous recessive for height (tt) we would arrange these two PARENT/P GENERATION plants around the outside of the Punnett square. And to find the possible genotypes made from this pairing you would fill the inside of the square with the corresponding allele from each parent. 

The offspring of the P-generation are called the FIRST FILIAL/F1 GENERATION. And the offspring of two F1-generation parents is known as the SECOND FILIAL/F2 GENERATION.