In breeding is any form of breeding that increases the likelihood of an animal being homozygous (having two identical genes as oposed to heterozygous different genes) for the same trait, as compared to an animal bred from unrelated parents.
Being homozygous for a trait can be good or bad, depending on the trait in question. If the trait is desired (say a particular color) then it may be good. Because a homozygous animal will always pass on the gene for the trait in question. In other words it will breed true. But some traits are bad. If you the animal is homozygous for a bad trait, it will always pass on the gene for the bad trait to it's offspring. Some traits are so bad that if an animal is homozygous for it, the animal will be sick or die.
Line breeding is inbreeding to "set" traits, so as to produce more consistant offspring. In other words, seeking to produce animals that are homozygous, and thus more likely to produce offspring that look like their parents. A "perfect" line bred animal would theoretically be homozygous for every trait, thus when bred to another "perfect" line bred animal of the same line, would result in a clone of the parents.
The conventional wisdom with line breeding is breed only ideal animals such that the offspring contain no more than 50% of the genetic material of a given animal. Since an animal is 50% from each parent, breedng two half siblings works - they end up with 50% of the genetic material from the shared ancester.
A x B -> C (50% A, 50% B)
B x D -> E (50% B, 50% D)
C x E -> F (50% C, 50% E = 50%B, 25%A, 25%D)
But a mother son cross would result in an animal that shares 75% of its genetics with its mother.
A x B -> C (50% A, 50% B)
B x C-> E (50% B, 50% C = 75%B, 25%A)
So if you have an exceptional animals that are half siblings, you might be justified in putting them together to try and generate similar offspring. But be prepared to cull anything that doesn't come out excellent.
Hope this helps.