Since the prefix co – means together or jointly, we can say that codominant genes are when

Genes have different variants, called alleles. Each individual possesses two alleles for each gene (one inherited from its father, and one from the individual's mother). Alleles can have different relations with each other, and some alleles have a stronger expression, dominating the determination of the trait in the individual.
In the case of codominance, both of the alleles are equally influential, so both of them are expressed in the individual.
The most famous example of codominance is in the A,B,O blood type alleles.

The A and B alleles are codominant, so when both of them are present in the individual, that individual expresses both A and B proteins, resulting in AB blood type.