A gelding is a male horse that has been castrated, removing the testes and the hormone-driven behavior of an intact stallion. Geldings are the most common riding horses: steadier in temperament, easy to keep in mixed groups, and free of the management a stallion demands. Castration is usually done while the horse is a young colt, but it can be performed at any age once both testicles have descended.
See also: the intact male a gelding was before castration, the young male usually castrated to become a gelding
Further reading: Gelding on Wikipedia; gonads and genital tract of horses from the Merck Veterinary Manual.