Shouts play a huge part in the story of Skyrim, and without giving too much away, they're what make your character special. Or rather, they're a symptom of why your character is special. Anyways, you'll find "word walls" as you explore, on which are incribed words of the Thu'um, an ancient form of magic that allows those who are practiced (or naturally gifted, as the case may be) to use their Voice to Shout into existence a variety of effects. The first Shout you'll learn, for example, is "Unrelenting Force", or the word "Force" in the Thu'um - by merely speaking it you blast an enemy with a wave of force. Such is the power of the Thu'um.
In total, there are eighteen Dragon Shouts to learn and master, all of which have their own special abilities. You will learn several for free just by playing through the main questline, but many will have to be found by locating the aforementioned "Word Walls". Even if you learn a word of the Thu'um, however, that's not enough to utilize it; you'll have to defeat specific foes - Dragons - to gain the power to expand your lexicon. This power naturally is Dragon Souls, which you'll automatically suck out of vanquished dragons. Many Shouts have three words to them, and each one will require one Dragon Soul to unlock. You can cast longer, more complicated versions of Shouts by hold down the appropriate button, while simply pressing the assigned button will quick-cast the basic version of the Shout.