The frog is the soft, V-shaped, rubbery structure on the underside of a horse hoof, pointing forward from the heels toward the toe. It cushions impact, aids traction, and helps pump blood back up the leg as the hoof flexes under weight. A healthy frog is firm and full; a shrunken or foul-smelling frog often signals thrush.
See also: the structure the frog is part of, trimming and frog health
Further reading: Frog (horse anatomy) on Wikipedia; disorders of the foot in horses in the Merck Veterinary Manual.