What does a heterozygous organism have?

It depends on the context. The only way to demonstrate practically that an organism is homozygous for a particular trait is to cross it (breed it) with another showing the recessive phenotype for that trait. When enough ofspring have been produced to be statistally significant then the conclusion can be drawn. It may also be possible to deduce whether an organism is homozygous by knowing the genotypes of the parents. Also, if you already know which phenotype is dominant then an organism showing the recessive phenotype is, by definition, homozygous. Of course, this all only applies to single gene inheritance!