Difficulty

Difficulty can be toggled via the Options menu in Minecraft. Changing this option has a direct impact in the gameplay itself. No setting impacts the quantity of hostile mobs that spawn, including peaceful. There are currently four difficulties in the game.

Hardcore difficulty
Hardcore Mode spawns you in Hard mode and does not allow the difficulty to change. If you die, the world is deleted.

Tips
If you are having trouble getting started or exploring caverns, the best thing to do is change the difficulty to "Peaceful". You will be able to explore caves or build a base/HQ much more easily. However, this does not give you the full experience, and you will have to turn the difficulty up if you want to get all the items and tools, such as the bow, or bones to make bone meal. With the recent 1.8 update, you may also find cobweb that, when destroyed with shears or a sword, give you a string. A good recommendation is to turn on Peaceful during the day, and then turn the difficulty back up at night IF you are prepared. This way, you can get the benefits of both ways to play.

Another idea is to change to peaceful whenever you venture into a cave, then turn it back up whenever you leave and go outside or to your house. That way you encounter the enemies above ground at night, but you don't have to worry about a creeper suddenly exploding in the darkness right in front of your face or a skeleton shooting at you from someplace you can't see when you go into a cave or dying from hunger by mining and jumping.

It can be difficult to get your feet if you are constantly attacked when first entering The Nether (particularly since some computers can freeze when switching between the two areas), so changing the difficulty to Peaceful will stop the Ghasts attacking you.

In the Nether, difficulty does not affect hostile mob spawning in any way, with the exception of hostile mobs not spawning at all in Peaceful.

Trivia

 * Notch made a tweet on October 22nd 2010, stating that he's "changing difficulty to realism. Lowest setting = creative. Highest setting = starve".
 * As of Beta 1.3, fewer passive mobs spawn on harder modes.