How far and fast a horse runs in one day really depends on the condition and breed of horse, as well as the terrain and weather conditions. That being said it is possible for some horses to travel as far as 60 miles or more in a day, but a more average distance is around 20-30 miles a day.
To put things into perspective, today in the Middle East, 26 mile horse marathons are won in just over an hour.
There are also longer marathon races called Modern Endurance Rides, which are 100-mile rides that are run by the same horse and rider in under 24 hours, but there is resting during the race for the horse.