A filly horse is a young female horse under 4 years old. Once a filly turns 4, she becomes known as a mare, even though fillies generally reach sexual maturity by 2 years of age.
Although fillies are sexually mature at age 2, they aren't bred until they become mares because they're still growing. A mare can breed once a year for several years. A horse used primarily for this purpose is called a broodmare. Once she gives birth to a foal, which is a baby horse, she nurses it for up to six months. A mare who is used for riding is also not bred regularly. In some rare cases a mare is fixed, becoming a spayed mare.
A colt is a young male horse under 4 years old. Once he comes of age, he is called a stallion. Stallions are often castrated for any of several reasons; two of the most common are that the stallion is unmanageable due to its testosterone levels and that it doesn't have desirable traits to pass on to young. Once castrated, a male horse becomes known as a gelding. On rare occasions a castration isn't complete, and the horse becomes a rigg. A rigg has some stallion character traits and in some cases is still fertile.