Basically, a punnet square is used to calculate the percentage chance that a certain trait will be passed on to the offspring.
For example, meiosis produces haploid cells, which contain half of the chromosomes of the parent. So, let us say that there are only two alleles (genes) that code for a certain trait. One for six fingers and one for five fingers (just for one hand). The allele for having six fingers is recessive (not dominant), and the allele for having five fingers is dominant (which means it will 'override' the other allele, resulting in five fingers even though the allele for six fingers is present).
Skipping over some irrelevant information, the haploid cells wouldn't be carrying both alleles. One would be carrying the allele for six fingers, and the other would be carring the allele for five fingers (let's say we're talking about sperm cells here, and that the mother has six fingers and has both of the recessive alleles). Basically, with a punnet square, we could calculate the percentage change of the offspring having five fingers (and vice versa).
Let's refer to the allele for five fingers as A, and the allele for six fingers as a.
A a
a a
That right there is a punnet square. It clearly needs some explaining, though. The letters represent the alleles, and the mother's alleles are on the bottom, and the father's are on the top (the other allele representations normally go on the left side of the square). The alleles are in the haploid cells, and the alleles were separated (because the chromosomes were also separated[you will understand better when you learn about meiosis]).
So, then, the letters actually also represent sperms and eggs (sperms are on the top and eggs are on the bottom in this example). Because of the way meiosis works, one sperm is carrying the dominant allele (which is for five fingers), and the other is carring the recessive allele (for six fingers). Both eggs are carrying the recessive alleles for six fingers. If the sperm carring the A allele fertilizes any egg, it will result in a child with five fingers. If the sperm carring the a allele fertilizes any egg, it will result in a child with six fingers.
How do we know which one will fertilize which? We don't! But we can calculate the percentage chance! Four sperms are produced from one cell in meoisis, so two are carrying the a allele, and the other two are carrying the A allele (all eggs are carrying the a allele). We can see here that there's a 50% chance that the A sperm will fertilze the egg, and there's a 50% chance that the a sperm will fertilize the egg. You might ask, "why?" Well, because A is dominant, so if it's present, it will cause 5 fingers, regardless of the fact that the allele for six fingers will also be present.