What determines whether an organism is heterozygous or homozygous dominant for a trait?

Two dominant alleles (such as AA) would make it homozygous dominant.  A dominant allele and a recessive allele (such as Aa) would make it heterozygous.  Homo recessive would be aa.

what is your question?
are you looking for an example of each?
or an explanation?

Heterozygous - Two different alleles for each trait.
          example - Bb.

Homozygous - Having two of the same alleles for each trait.
          Homozygous Dominant - having two dominant alleles for each                
          trait.  
                example - BB
          Homozygous Recessive - having two recessive alleles for each
          trait.
                example - bb


What determines if an offspring is heterozygous or homozygous depends on the parents' genotypes.
Make a punnett square to determine whether the offspring will be heterozygous or homozygous dominant for a certain trait.
There may be various possibilities, so you may have to use the phenotype to determine the genotype.

Genes generally occur in pairs...in humans we inherit one gene from our mothers and one from our fathers. If we get the same allele (variation of the gene) from each parent, then we are homozygous for that gene. If get a different allele from each parent, then we are heterozygous.