Saddles are needed to ride dragons.
Saddling your DragonEdit
To place a saddle on your dragon, hold the saddle in your hand and right-click your dragon. If the dragon already has a normal saddle on, a saddlebag will replace the normal saddle.
There does not seem to be a way to remove a saddle from a dragon once it is on.
A regular saddle will have the standard seat size and two straps around the body of the dragon.
A saddlebag has a longer seat and three straps around the body of the dragon.
To make a saddlebag, combine a saddle and a chest in a crafting table.
If your dragon is wearing a saddlebag, you can access the inventory by riding the dragon and hitting the "R" key while on the dragon. It has a 27 item slot inventory and 4 additional slots for the dragon's armor.
There is a display for the Health % and Armor Value of the armor the dragon is wearing, if any.