A horse's sire is simply his or her father. The term is archaic when used of humans (you may have seen it show up in fantasy novels, although the more common usage there is as a term of address for a king).
The word might also be used to refer to the stallion himself. A stallion might be advertised as "Sire of multiple Futurity winners," for example, or as the sire of a particularly well-reputed horse. And it's used as a verb "Sires more fillies than colts" (Often a selling point used by stallion owners).