Dragon shouts are special attacks, also known as "Thu'um" to the Nords which are learned from word walls.
In order to learn a shout you need to take words from word walls and infuse them with dragon souls which have been absorbed from dragons you have killed.
There are 20 Shouts total which can be learned over the course of the game.
Press and hold 
(or RB for console) to use a shout. The longer the button is held, the more words of the shout are used (if you know more than one).
After using a shout, there is a small delay before you can use another. This is indicated by a glow on your compass: when the compass returns to normal, you can shout again.
Using a shout in a city a guard will come up to you and tell you to stop with that shouting and you're making them nervous if you refuse you won't get a bounty but he will tell you he has his eye on you
Each Shout is made up of three words. You must learn these words by finding Word Walls scattered throughout the world. After learning a word, you must unlock it with a dragon soul before you can use it. Dragon souls are absorbed by slaying dragons.