The difference between heterozygous and homozygous individuals is that heterozygous beings possess a genotype that carries differences within it, whereas a homozygous has no differences of this nature. A good example of this is human blood type, where an AB blood type is heterozygous, and an O blood type is homozygous. One possesses both the A and B alleles, and the other only possesses the O alleles.
An allele is a genetic trait that provides a small difference between several examples of a species, such as skin pigmentation or blood type. Most of the time, this difference is so small as to be invisible without scientific tools. In the case of blood type, microscopes or chemical tools are required to determine which blood type an individual person has.
The reason a gene can have two different alleles in a single individual is because most multicellular organisms, like plants and animals, have two sets of chromosomes. This allows the possession of two different copies of the same gene in an individual, resulting in small genetic variances within many examples of a species. Heterozygous individuals are those who have two genes that are different, like in the case of the AB blood type.