What determines our naming of genes as dominant or recessive?

Genes are particular parts of DNA that code for proteins, they are not dominant or recessive. Most genes have alternative variants called alleles, e.g. in the ABO blood system there are three alleles, which code for the red blood cell proteins A or B or no protein O.

A dominant allele is one that is expressed whenever it is present, so if you have an A allele in your ABO gene, you will have an A antigen on the surface of your red blood cells.

If an allele is recessive, both copies must be the same recessive allele to be expressed, so O is recessive, only 2 O alleles will lead to no antigen on blood cells, in combination with either A or B, the existence of A or B antigens will prevent the O type being expressed.

i'm uncertain what you're question potential. i'm guessing the two styles of genes are dominant and recessive. to respond to your different questions, the genotype is the genetic makeup of an organism. we won't see what a individual's genotype is. what's expressed and considered is the phenotype. we are able to infer what a genotype relies on the phenotype. If the genotype is two recessive alleles, then the recessive phenotype would be considered and we are able to be responsive to what the genotype is. If we see a dominant phenotype, then we are able to infer that the genotype will the two be homozygous dominant or heterozygous.