In Evony, a stable is a building where your horsemen (cavalry, cataphracts) are kept. Stables decrease the time it takes to train horsemen, and the higher level your stable, the faster your horsemen will move.
You cannot research Horseback Riding technology without a stable and you cannot build a Relief Station without Horseback Riding. Horseback Riding and your Relief Station are vital to any city, or else you’ll be waiting a day or two to attack a player who’s far away.