Genotypes can be purchased at the Geneticist via the Racing tab. These will appear on the Genetics page below the table for ratings. It will look like a long string of letters above a long string of colored bars. Essentially, you are looking at the exact genetic code for your horse! Genotypes must be purchased for a stud to stand publicly.
Genotypes give specific genes for the following genetics (in order):
Color: Extension, Agouti, Grey
Surface/Distance: Dirt, Turf, Speed, Stamina
Performance Genes: Development, Longevity (Health), Temperament, Conformation, Intelligence
They will be one of three colors: green, yellow, or red. Red genes are xx genes, as you can tell if you compare it to the list of letters above the bars. Yellow genes are Xx genes, and Green genes are XX genes. Generally speaking, the second half of the genotype (D to I) is the most important part, and is where you want to look for more green and less red. Fertility, the last gene on the first strand, is also of moderate importance.
Colors are not very important, unless you are breeding for specific colors. Likewise, surface and distance aren't bad if they are red - but you do want to check them to make sure you are breeding compatible horses!