A true-breeding organism, sometimes also called a purebred, is an organism that always passes down certain phenotypic traits (i.e. physically expressed traits) to its offspring. An organism is referred to as true breeding for each trait to which this applies, and the term "true-breeding" is also used to describe individual genetic traits.
In Mendelian genetics, this means that an organism must be homozygous for every trait for which it is considered true breeding; that is, the pairs of alleles that express a given trait are the same. In a purebred strain or breed, the goal is that the organism will "breed true" for the breed-relevant traits.
Apomixis and parthenogenesis, types of asexual reproduction, also result in true breeding, although the organisms are usually not homozygous.
Examples
A purebred variety of cat, such as Siamese, only produces kittens with Siamese characteristics because their ancestors were inbred until they were homozygous for all of the genes that produce the physical characteristics and temperament associated with the Siamese breed.