Taming a horse is actually quite easy. They are found in grassy plain biomes but if you want to skip looking for one quickly create a new world and use the seed "horses". When you spawn in run forward a short time and there should be 3-5 horses in the plains ahead.
To tame one, approach a horse and tap "Ride" on the bottom of your screen. Chances are you'll be thrown off almost instantly. The beauty of taming a horse, however, is to just keep on trying to ride it until you don't get thrown off. You'll know the horse is tamed by the hearts that appear over its head.
Although unrelated to the achievement, if you wish to control your newly tamed horse, you will need to find a saddle and throw it on him.
From ganster903: "if you feed a horse apples, it lowers to amount of times you need to get on the horse before it's tamed."