Minecraft – How to get a horse/horse saddle

In short:

1. Right-click on the horse and you will mount it.
2. After a short while you will be thrown off
3. Repeat until the heart animation shows
4. You have now tamed the horse and can saddle it.

It will usually take 3-4 attempts to tame the horse, but it can take as little as 1. To speed up taming, feed the horse before trying to mount it. The following foods boost the likelihood of taming:

Sugar         | +3% 
Wheat         | +3% 
Apple         | +3% 
Golden Carrot | +5% 
Bread         | +3% 
Golden Apple  | +10%

From the wiki:

Adult normal horses, donkeys, and mules can be tamed; foals, skeleton horses, and undead horses cannot. It is necessary to tame a horse in order to breed it, give it equipment, or ride it for any length of time.

To tame a wild horse, approach it and right click on it with an empty hand. You will climb on top of it, and will likely be thrown off. After a few attempts, it will allow you to remain on it and will give off heart animations. At this point you can dismount it by pressing the sneak key and place a saddle on it, or you can place a saddle without using the sneak key at all by right-clicking on the horse with the saddle in hand. Horses can also be saddled by opening your inventory while mounted and moving the saddle to the appropriate slot in the horse's inventory.

The chance of successful taming on the first try is almost 0%, yet it can be done. Taming on the first attempt is very rare. After the first attempt without taming the horse, the chances increase by about 5%, and usually one can then tame a horse after about 3-4 attempts. Taming can be sped up by feeding the horse any of the approved foods when you are standing next to it - see column "Taming prob." in the table below.

One of the more efficient ways to try and tame a horse is to build walls around it, making the space within no more than 10x10, don't let the horse have too much space. Should you fail to tame the horse, you can always come back later, provided the enclosure contains the horse (i.e. don't put blocks near the edge, etc.). Another tactic is to use water, since it will limit the horses ability to move, allowing you a better chance to tame it.

You can attach a rope leash to it and guide it home that way. A leash is made with string and slime, as follows:

If you need to take a break, you can tether the horse to a fence. This allows you to wander off and do whatever you need to do. Provided you remember where he/she is, you can return and get the horse back later.

More information can be found on the wiki.