Homozygousmeans that the organism has two copies of the same allele for a gene. An organism can be homozygous dominant, if it carries two copies of the same dominant allele, or homozygous recessive, if it carries two copies of the same recessive allele.
Heterozygous means that an organism has two different alleles of a gene. For example, pea plants can have red flowers and either be homozygous dominant (red-red), or heterozygous (red-white). If they have white flowers, then they are homozygous recessive (white-white). Carriers are always heterozygous.