I've read that a professional photographer should have at least two camera bodies, with one serving as a backup. How important is it to have a backup camera body? Consider both amateur and professional use cases.
Short answer: Are you mad ? !!!!!!! [ :-) ]
Longer answer: For a professional lack of a backup body is ~= "death-deferred".
You could consider that "being able to access an alternative acceptably quickly at an acceptable cost" is the equivalent to having a backup body, so if you were a studio only photographer and there was a 24/7/365 camera hire shop on the ground floor who always had acceptable bodies in stock to hire then, maybe that would do, but also, pigs may fly.
If you are photographing a wedding / sports event / social function / wild-life assignment / ... and "my camera broke" is not going to be acceptable, and is liable to cause immense (or any) embarrassment, financial loss, loss of prestige and reputation, reduction or elimination of future opportunities, ... - then you must have one. If you are a free lancer and nobody cares if you front with photos and it doesn't bother you either, feel free to trust Murphy. Most professionals are not in that position.
Many professionals who carry a second camera will use it in parallel with their "main camera", allowing changing camera rater than changing lenses. Or more options when changing.
A backup body CAN be whatever you "can get away with". But when you consider the cost of lenses, flashes, memory cards, computer equipment, software, general accessories and your costs of staying alive, then the cost of an acceptably good backup body becomesa small fraction of the total.
Note: Try not to utterly depend on having to use medium format, or Leica or ... :-).
If you are serious, it's nice, and the cost of an "acceptable enough" second body can be quite modest. When you buy an entry level used camera as your first DSLR and it stretches your budget, then buying two is obviously not an option. But if you carry a D800, then adding a used D7000, or not selling it when you upgrade, may make a lot of difference to your life. It also depends on what sort of photography you do, where you go to take photos and how much you care.
I consider myself "semi professional" in that my "day job" is as an electronic designer, but photography is an obsession (just ask my wife) and I much more than pay for my camera equipment with paying work, (assuming you ignore the hourly rate for my time after capital costs :-) ). I have had a camera die due to being dropped (a very rare occurrence) at midnight on returning home from a paid stage show when I had an unpaying wedding that coming morning. The backup was inferior but did the job.
I have travelled to China on numerous occasions in recent years on business. I can make do with a combined pocketable video/still camera for business (Sanyo XACTI) but take a DSLR (x2) for frenetic running around outside factory hours and for tourist mode activities after business phase ends. (eg the Xian visit that produced the bronze horse photo on the front page at present was amateur only on the Shanghai-Xian-Shanghai portion. I have 1000+ photos from the Terracotta warriors. I would hav estill had them if 1 of my 2 DSLRs had died. Neither did.) So the DSLRs are used for business as I have them there, and give me better business photos, but are "essential" for amateur use. I have had a camera become highly walking wounded while in China (I live in NZ). The backup I carried was as good image wise, just not as many bells and whistles. I do more unpaid or expenses only weddings/parties/events etc than paid ones. Both sorts are fun, but one sort I get paid for having fun. I have never had a camera fail at a wedding or event or stage show during the actual event, but I carry two cameras with a specialist lens on the second. (Usually either a 50 mm prime or a 500mm mirror lens on the backup depending on circumstances. ie 500mm is not usually much use at a wedding during a church service (has happened) and a 50mm prime is less useful on a long rally straight).
Conclusion in both cases:
You'll never regret having a second camera available at a moment's notice after you forget what you paid for it.
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!
The professional case is fairly obvious, I would think. Photo shoots can be expensive and time consuming to set up. It could be limited location options, assistants, model availability, and more that is a factor here, but it boils down to the fact that a faulting camera with no back up means a lost shoot and that can be a lot of money.
The other professional case could be around things like wedding work. In the one and only wedding I've shot, I used dual cameras with differing focal length ranges with high grade zooms (all f/2.8 constant) to give me complete shot options without a lens swap. Kind of handy when you have a moment or two to make the shot.
For amateur, the case isn't very strong unless you can afford it. When you can, it's nice for travel and nature work. I often, again, sling the dual camera where one is set up with a wide angle or a macro prime and the other is set up with a telephoto option. Makes it easy, as with the wedding shoot, to get the quick shot away with high quality and no lens swap. Of course, you can go super zoom, but you trade quality of glass, and thus image quality, for that convenience.
At any rate, I don't mind carrying two K-5s, they're nice and compact. :)
Professional get paid when they go to a shoot, and if the camera fails, that $1000 job would become a $0 job + one very pissed off client. So it is a MUST.
For people not getting paid doing photography however, having a camera failure really doesn't have that big an impact.
It really boils down to how important is the shoot. For example it is your sister's wedding day, you would hope that you have a backup camera for such a once-in-a-lifetime event.
For things like taking photographs of your pet in a park, it simply doesn't matter, you can go another day and take some more photos when your camera is working again.
However, for most of the time, when not doing professional jobs, a simple point and shoot would be a good enough backup. If you absolute demand DSLR quality getting a second hand DSLR for cheap would do a perfect job, too.
Say you saved up good money and went on a holiday with your loved ones, and you both enjoy taking a lot of photos. Then a backup camera would make a lot of sense. For my travels I take my 60D and a large aperture point-and-shoot. My girlfriend would happily use the point and shoot (she enjoys photography too and she knows how to operate a DSLR well). In case we drop any one of them in water, we will still have one camera left for capturing our lovely vacation.
Personally, I would have a backup, however that would be a good quality point and shoot, not a DSLR. For lighter weight and versatility. Almost all do macro, and some have amazing zoom range, they are also cheap :D
Hope this helps.