If an horse is heterozygous for a trait, that means they possess both the dominant and recessive gene, whereas a horse that is homozygous for a trait, that means it only carries either the dominant or recessive gene.
Implications for breeding
If "A" is dominant and "a" is recessive, then only the offspring that is homozygous recessive will have the recessive phenotype, AA and Aa will display the dominant phenotype.
If two heterozygous horses are bred together there is a 3 possible combinations the offspring could end up with. If sire is Aa and dam is Aa, the offspring will be one of AA, Aa, or aa.
If a heterozygous horse and a homozygous horse are bred together, the offspring will have either Aa or AA, if the homozygous horse is homozygous dominant and Aa or aa if the horse is homozygous recessive.
If two homozygous recessive horses are bred together, the offspring is guaranteed to be homozygous recessive as well. If two homozygous dominant horses are bred together, the offspring is guaranteed to be homozygous dominant as well. If a homozygous dominant horse and a homozygous recessive horse are bred together, the offspring are guaranteed to be heterozygous.