THE LEGEND OF ZELDA Links horse's name is Epona
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Epona is the canonical name of Link's horse, but in Twilight Princess and Breath of the Wild, the player can name their horse manually.
The Horse Update
Epona is the traditional name of the horse, such as in Majora's Mask and Ocarina of Time. However, Twilight Princess gives you the option to rename your horse. While the answer above is true for Link's horse, throughout the series the name of Princess Zelda's horse has never been official; however in 'the Legend of Zelda' by Valiant comics Zelda's horse was named 'Storm'. This however is not considered canon.
you can get horses by feeding them apples and taming them,it will give you a name choice for your new horse and then you have a new horse!
firstly you need to tame your horse, there are many tutorials for this, next you have to obtain a name tag from a dungeon test. these are not to rare in the new update. then you go over to your horse right click it with the name tag and you should be prompted to put a name in a box and then when you come out of the menu you will be able to see the horses name if you're a few blocks away from him and looking directly at him.
There is ponyta which evolves into rapidash.