DrZhark's Magic Carpet Mod is a mod that adds flying carpets and colored rugs into Minecraft. They can be used as decorations, and enchanted (once combined in your inventory) to become magical flying carpets. Grab it from here. For more mods, be sure to check out our list of best Minecraft mods.
By default, rugs can be placed in the world rolled up. You can unroll them by left-clicking on them. When combined with a Bottle of Enchanting, a Blaze Rod, or an Enchanted Book, they can be turned into a Magic Carpet with which you can fly around the world on.
Riding the Magic Carpet is fairly easy, as you can right click it to mount it. It works the way you'd expect a magic carpet to work, by either diving or lifting, and flying forward.
You can watch the whole mod in action through this video by CaptainSparklez, seen below: