As of right now, at least two new shouts have been found in the Dragonborn DLC and you can gain most of the words from the main quest.
The first shout is Bend Will which allows you to temporarily tame creatures. The second word allows the control of people and the third word allows control of Dragons. When you use all three words, you can tame a dragon. When it lands, you can then mount the dragon and fly on it. You won't have control of the flight of the dragon but you can use it to fight. While flying, you can use magic spells or use the dragon's breathe as an attack.
The second shout grants you greater power in general. You become tougher, stronger, and your shouts are amplified. You'll gain two of the words over the course of the main quest but the third has to be find during a side quest.
