A homozygous trait is when two of the same kind of alleles combine to form a trait. Think of this as traits in common in two different creatures which will continue in their offspring.
A good example are a very tall man and a very tall woman who have a child. This child grows into an adult and is very tall as well. It could be said that the child's height is a homozygous trait which he got from both parents.
The same could be said if the child had two short parents and the child was short when it reached adulthood. The shortness in the child would be a homozygous trait adopted from his short parents.
Heterozygous is a trait when two different kinds of alleles combine to form a trait. When this occurs there is a dominant and a regressive allele in the pair, and the dominant allele will indicate which trait the child will get.
Take the previous example from where two parents have a child. But in this example one parent is tall and the other is short. The allele for shortness is dominant and the allele for tallness is recessive. The child, when he reaches adulthood, is also short, indicating that the allele for shortness overrode the allele for tallness. This would be a heterozygous trait.