A true breeding is a kind of breeding wherein the parents would produce offspring that would carry the same phenotype. This means that the parents are homozygous for every trait. An example of true breeding is that of the Aberdeen Angus cattle. The cattle have been black for many generations, which is an indication that such trait passed on from many generations remains pure. Another example is the mating between two dogs of the same breed. The traits of the resulting offspring would therefore be more predictable.
With plants, true breeding occurs when plants produce only offspring of the same variety when they self-pollinate. For instance, a plant that has blue flowers will produce only seeds that will grow into plants that have blue flowers. With true breeding, the trait is passed on to all subsequent generations. For this to occur the parents are homozygous for a trait -- which means the parents must be both dominant or both recessive.
True breeding, however, implicates a limited gene pool. As such, there is a high tendency of a particular trait to be inherited (e.g. genetic disorders) that could potentially be detrimental to the health of the offspring.