The genetics of blood type is a relatively simple case of one locus Mendelian genetics—albeit with three alleles segregating instead of the usual two (Genetics of ABO Blood Types).
Eye color is more complicated because there's more than one locus that contributes to the color of your eyes. In this posting I'll describe the basic genetics of eye color based on two different loci. This is a standard explanation of eye color but, as we'll see later on, it doesn't explain the whole story. Let's just think of it as a convenient way to introduce the concept of independent segregation at two loci. Variation in eye color is only significant in people of European descent.
At one locus (site=gene) there are two different alleles segregating: the B allele confers brown eye color and the recessive b allele gives rise to blue eye color. At the other locus (gene) there are also two alleles: G for green or hazel eyes and g for lighter colored eyes.
The B allele will always make brown eyes regardless of what allele is present at the other locus. In other words, B is dominant over G. In order to have true blue eyes your genotype must be bbgg. If you are homozygous for the B alleles, your eyes will be darker than if you are heterozygous and if you are homozygous for the G allele, in the absence of B, then your eyes will be darker (more hazel) that if you have one one G allele.
Here's the Punnett Square matrix for a cross between two parents who are heterozygous at both alleles. This covers all the possibilities. In two-factor crosses we need to distinguish between the alleles at each locus so I've inserted a backslash (/) between the two genes to make the distinction clear. The alleles at each locus are on separate chromosomes so they segregate independently.*
As with the ABO blood groups, the possibilities along the left-hand side and at the top represent the genotypes of sperm and eggs. Each of these gamete cells will carry a single copy of the Bb alleles on one chromosome and a single copy of the Gg alleles on another chromosome.
Since there are four possible genotypes at each locus, there are sixteen possible combinations of alleles at the two loci combined. All possibilities are equally probable. The tricky part is determining the phenotype (eye color) for each of the possibilities.
According to the standard explanation, the BBGG genotype will usually result in very dark brown eyes and the bbgg genotype will usually result in very blue-gray eyes. See the examples in the eye chart at the lower-right and upper-left respectively. The combination bbGG will give rise to very green/hazel eyes. The exact color can vary so that sometimes bbGG individuals may have brown eyes and sometimes their eyes may look quite blue. (Again, this is according to the simple two-factor model.)
The relationship between genotype and phenotype is called penetrance. If the genotype always predicts the exact phenotpye then the penetrance is high. In the case of eye color we see incomplete penetrance because eye color can vary considerably for a given genotype. There are two main causes of incomplete penetrance; genetic and environmental. Both of them are playing a role in eye color. There are other genes that influence the phenotype and the final color also depends on the environment. (Eye color can change during your lifetime.)
One of the most puzzling aspects of eye color genetics is accounting for the birth of brown-eyed children to blue-eyed parents. This is a real phenomenon and not just a case of mistaken fatherhood. Based on the simple two-factor model, we can guess that the parents in this case are probably bbGg with a shift toward the lighter side of a light hazel eye color. The child is bbGG where the presence of two G alleles will confer a brown eye color under some circumstances.
*If the two genes were on the same chromosome this assumption might be invalid because the two alleles on the same chromosome (e.g., B + g) would tend to segregate together. Linked genes don't obey Mendel's Laws and this is called linkage disequilibrium.