[H]ow exactly we mark out genotype in science? Do we use D's and r's?
General rule
The general rule is to indicate the dominant allele with a upper-case letter and the recessive allele with a lower-case letter.
The most commonly used letter is the first letter of the alphabet. The two possible alleles are a and A and the four possible genotypes here are therefore aa, aA, Aa and AA. Only aa has blue eyes in your example. Note that it is very common to merge together Aa and aA (that is to not consider whether the A allele comeswas inherited from the father or from the mother) and just call it Aa.
More than 2 alleles OR no clear dominance relationship
It is quite common to use A1 and A2 as well but when doing so, we don't implicitly indicate a relationship of dominance. This notation also has the advantage to be able to deal with cases where there are more than two alleles segregating in the population (A1,A2,A3). Note that in reality perfect dominance is quite rare.
More than 1 locus
When dealing with more than one locus, we generally use A/a for the first locus, B/b for the second locus, C/c for the third etc... However, in general when dealing with several loci, authors don't have identical loci ad therefore they directly name them and use appropriate letters. For example, in models of recombination under control of a modifier locus, the modifier locus can often take the values M and m, while a random linked locus under purifying selection would take the letters A and a.