I love to ride horses in video games, so when I found out you could ride a horse in Minecraft, I had to give it a try. I talk about this some in my instructable about Minecraft Animals/Mobs, but I wanted to devote an Instructable to how to (tame and) ride a horse in Minecraft.
Step 1: What You Need
In order to ride a Horse, all you need is a Saddle. Unfortunately, you cannot make a saddle, you have to find one. You can find them occasionally in chests around the world.
Optionally, you can put Horse Armor on Horses (not Donkey's or Mules). This also can only be found, not crafted.
You can equip a Donkey or Mule with a Chest to have it hold supplies for you (you cannot add a chest to a Horse).
Step 2: Taming
Before you can ride a Horse, you need to tame it.
The way to tame it is, while not holding an item, click on the Horse like you would to use an item. The Horse will most likely buck you off. You need to repeat this until hearts appear, but you can also feed the Horse Apples, Wheat, Golden Apples, Golden Carrots, Wheat, or a Hay Bale to help tame it.
Now your Horse is tamed.
Step 3: Riding Your Horse
Once your Horse is tamed, it will no longer buck you off, but you need to put a saddle on it before you can control where it goes.
Either get on the Horse or point towards the Horse and open your inventory menu. This will open up your inventory and the Horse's menu. Drag the saddle up where the outline of the saddle is. Now you can ride.
Optionally, put armor on your Horse. It goes right below where you put the saddle.
You cannot ride in water that is deeper than 1 block. If you do, you will be thrown off the Horse and the Horse will start drowning. If you do this, use a lead to get your Horse out of the water so you can ride it again.
A horse can walk over flat land and over blocks that are 1 block high. I've read online that they can jump high, but I've only been able to get mine to jump 2 blocks high. To do this you need to hold down the space bar until the action bar (above your items) reaches the top of the blue (see image). Then let go and your horse will jump. Don't forget to go forward at the same time.
Step 4: Chests
You can only use a chest on a Donkey or a Mule. To equip it, just use it on the Donkey or Mule just like you would use any other item.
Unlike saddles and Horse Armor, you cannot get the chest back from the Donkey or Mule. The only way to get it is to kill the Donkey.