Linebreeding is defined by the same source as: "the interbreeding of individuals within a particular line of descent usually to perpetuate desirable characters"

As you can see, the two definitions are virtually identical. The practice of linebreeding animals is essentially inbreeding. If you hear people talk about dogs being "tightly line bred", "closely line bred", or having "a high in-breeding/line-breeding coefficiency", it generally means "inbred". (Not that there's anything wrong with that term, but it upsets many folks' human sensitivities to call what dog breeders do "inbreeding", so the term "line breeding" was developed instead. So, when most breeders talk about line breeding, they are usually talking about inbreeding the animals to some degree or another.

First a bit of history for context: Marrying one's human relatives to a close degree of consanguinity was and still is illegal/taboo. To what degree the relationship is prohibited varies by state law. Most of these laws came about because before much was known about genetics and passing on of genes or characteristics, it was discovered that certain close relatives' offspring often had disturbing defects or characteristics. Hence, the prohibition against incest arose.

In animals, however, no such legal prohibitions exist, and breeders of various species have experimented with varying degrees of linebreeding/inbreeding for centuries. Although one can see some disturbing genetic flaws rise to the top of the gene pool once in a while with uninformed inbreeding, the flip side of the coin is that the very best characteristics can also be "fixed" in a line of animals by doing that very thing. Most of the prepotent animals (studs and dams) who consistently produce animals that look alike, act alike, hunt alike, etc., are inbred to a certain degree, and have these genes "fixed" in them so that they routinely pass them on to their offspring. Other animals, who are themselves very nice specimens of their breed, sometimes never produce any offspring of any significance (in terms of memorability, performance, etc). Many times these animals are the product of outcrosses (breeding of two basically unrelated animals), and thus don't pass on any of their better characteristics to their offspring.

The "breeds" of domesticated canines that we have in existence today are all products of such inbreeding programs. Man looked at wolf, for instance, and liked the way he stalked his prey, pausing briefly before springing in for the kill. He decided he'd like to have a dog like that work for him. So, he started feeding the wolves scraps to keep them hanging around the fires/caves. He probably fed the ones most that were of the most use to him and had the characteristics he most favored, and the ones who were most receptive to his attention. In time, those reproduced with one another around his camp, and the offspring probably shared those characteristics. The ones that didn't were likely driven off. Eventually, conscious decisions were probably made about which "lines" to perpetuate through intentional breedings, and "soon" there was a dog for every purpose under the sun (e.g. herding, hunting, carrying packs, pulling sleds, etc.), and some for no purpose at all (i.e. lap dogs as more luxury and leisure time became available).