Extremely important for professionals. Most magazines publish photographs, not excuses. So if your camera gets attacked by a bear, falls into a lake, gets stolen, you still have to be able to bring back photos. Many events are once in a lifetime or occur extremely infrequently, which gives you only one chance to capture them.

For amateurs there is absolutely an advantage too. Yet, I wouldn't call it so absolutely essential. You may only go to Borneo (substitute your own far-off dream destination here) or see a total solar eclipse once in your life but even if you so not bring back any photos, you will have your memories for a while at least.

Note that everyone speaks of a backup camera but you also need backups for everything else. Many professionals have experienced failed lenses (including myself) which is crippling. With digital cameras there are multiple single-points-of-failure too. Consider what happens if your charger breaks after your set of batteries goes empty. That is why it is easier to go with identical or very similar backups. For me a K-5 + K-7 works but the K-7 + K20D was a nightmare. Another backup solution which I use is to have an ultra-zoom, this saves weight and works well for non-action and non-low-light photography.

Should you travel without a backup, you can limit your losses by downloading frequently and sending images away. Everytime I get 4GB of photos, I burn them to DVD twice. One copy is kept with me and the other is mailed to myself. This limits the loss of photos already taken but not potential future photos. DVD or other optical media are not only virtually indestructible, they also have no value to potential thieves, unlike portable hard-drives which are popular.

For event photography, you will often see second non-backup bodies to shoot with more than one lens quickly. Those photographers usually also have a backup too. In the some cases, like weddings even, professionals can use backup photographers too!