Terminology
The male parent of a horse, a stallion, is commonly known as the sire and the female parent, the mare, is called the dam. Both are genetically important, as each parent provides half of the genetic makeup of the ensuing offspring, called a foal. (Contrary to popular misuse, the word "colt" refers to a young male horse only; "filly" is a young female.) Though many horse owners may simply breed a family mare to a local stallion in order to produce a companion animal, most professional breeders use selective breeding to produce individuals of a given phenotype, or breed. Alternatively, a breeder could, using individuals of differing phenotypes, create a new breed with specific characteristics.
A horse is "bred" where it is foaled (born). Thus a foal conceived in England but foaled in the United States is regarded as being bred in the US. In some cases, most notably in the Thoroughbred breeding industry, American- and Canadian-bred horses may also be described by the state or province in which they are foaled. Some breeds denote the country, or state, where conception took place as the origin of the foal.
Similarly, the "breeder", is the person who owned or leased the mare at the time of foaling. That individual may not have had anything to do with the mating of the mare. It's important to review each breed registry's rules to determine which applies to any specific foal.
In the horse breeding industry, the term "half-brother" or "half-sister" only describes horses which have the same dam, but different sires. Horses with the same sire but different dams are simply said to be "by the same sire", and no sibling relationship is implied. "Full" (or "own") siblings have both the same dam and the same sire. The terms paternal half-sibling, and maternal half-sibling are also often used. Three-quarter siblings are horses out of the same dam, and are by sires that are either half-brothers (i.e. same dam) or who are by the same sire.
Thoroughbreds and Arabians are also classified through the "distaff" or direct female line, known as their "family" or "tail female" line, tracing back to their taproot foundation bloodstock or the beginning of their respective stud books. The female line of descent always appears at the bottom of a tabulated pedigree and is therefore often known as the bottom line.
"Linebreeding" technically is the duplication of fourth generation or more distant ancestors. However, the term is often used more loosely, describing horses with duplication of ancestors closer than the fourth generation. It also is sometimes used as a euphemism for the practice of inbreeding, a practice that is generally frowned upon by horse breeders, though used by some in an attempt to fix certain traits.
Read more about this topic: Horse Breeding