The term homozygous is used when two alleles of the same gene are equal. The name is derived from the Latin word homo = equal. The opposite term is heterozygous, where hetero = different, meaning the two alleles of the same gene are different.
The grey gene has two known alleles: G and g, with the capital being dominant over the lower cased letter. G is the allele that turns a horse's coat white while g is basically the 'off' switch that leaves the coat unaffected. Though genes can have multiple functions, turning a coat white is not one of g's functions, meaning it does not influence it.
Every horse inherits one allele from each of its parents. If our horse has inherited a G from its mother, and a g from its father, it has the Gg genotype. When the two alleles inherited from both parents are equal, we say the horse is homozygous for the gene/colour/characteristic. When the two alleles are different, then we say the horse is heterozygous.
Below are several examples:
Horse 1: g x g = gg = homozygous
Horse 2: G x g = Gg = heterozygous
Horse 3: g x G = gG = heterozygous (often, we place the capital letter first as the result is the same)
Horse 4: G x G = GG = homozygous
As you can see, it does not matter which alleles a horse has, nor what the allele's dominance status is. When they're the same, we always call them homozygous. When they're different, they're called heterozygous.
Note: a horse homozygous for a particular trait, for example for the G gene, will always pass that allele on to its foals. This means that a particular colour may spread fast in a population, if, for example, a very popular stallion is homozygous for it.
If the horse is heterozygous for the colour, it will pass on each of its alleles with a 50% chance. A stallion with Gg will pass on its G for 50% of the time, while the other half of the foals will receive a g. If we do not take the mare's genetics in account, that roughly means half of the stallion's foals will be grey (they have inherited G) and half will be anything but grey (they've inherited g).