A flip-flop is the American/UK term for a flat-soled sandal with no heel or ankle strap. It is a simple, flat sole with a Y-shaped strap, with the leg of the Y emerging from between the big toe and the second toe, and the two arms attaching to the sole on either side of the foot. In Australia they are called thongs, and in many equatorial countries they are called slippers. There are even rumors that some call them by a more onomatopoeically proper term of 'slip-slaps', although in what region of the world this might happen I have not be able to find.
I have always worn flip-flops in any context that I could not go barefoot; they are the closest thing to a truly multipurpose footwear that the world has yet found. I would not be espousing them on my word alone, however. I had for a long time supposed that this was simply personal preference on my part, until I joined the Peace Corps, and moved to a country where flip-flops were the footgear of choice. Traveling to other equatorial countries, I found this to be true time and time again. With the popular opinion of entire nations behind me, then, I will node unto you, the one true footgear.
Now admittedly, there are a number of downsides to flip-flops. Obviously, they aren't very good in the snow and the ice, and they give no protection from crushing or other trauma from above. They are hard to sprint in, and rather hard to run in at all until you become used to them. The basic model will also fail in wet conditions (as you will find you suddenly have no traction between your foot and the sole), and may potentially be a problem for those with podiatric problems, as most flip-flops give no arch support nor much cushioning for shock absorption. The astute shopper can find flip-flops that have both water-ready surfaces and arch-support and added cushioning. Ironically, although they were originally marketed as beach shoes, and remain ridiculously popular amongst surfers, flip-flops are hard to walk in when sinking into deep sand.
On the plus side, they are wonderful for hiking and walking; the wonder of flip-flops is that they have only three rubbing points that contact with your foot, and if you select your pair correctly, you will probably never have a blister. If you do get blisters the first time you hike in flip-flops, it will probably be the last time you get blisters -- ever. Unlike other forms of footgear, one set of callouses will function to protect you from getting blisters from any other pair of flip-flops that you might buy. Breaking in a new pair of flip-flops is infinitely less painful than a pair of hiking boots. They also have the benefit of being easy to take off if you are confronted with a cliff face or stream (you do climb barefoot, don't you? It's much quicker and easier).
Perhaps the greatest benefit, however, is hygiene. It is nearly impossible to cultivate any sort of foot fungus if you wear flip-flops as your primary footwear. These infections prefer dark, damp, warm habitats, and the flip-flop provides none of these. Ironically, flip-flops are often touted as a hygiene-aid because they can be worn in public showers, thus preventing contact with infected floors and drains; if only the other patrons of the bathhouse wore flip-flops more routinely, this would no longer be a problem.
There a couple of common myths about flip-flops that should be shot down. First, many people believe that flip-flops are bad for your posture. This is not true; in fact, "zero heel inclination" is the baseline used in physiologic studies on posture, and walking barefoot (or with no heel) is exactly what your body was designed to do. As it happens, your body will automatically adjust to most heels, although there is a point at which a heel will be too high for your body to adjust to properly1. One study2 focusing on tension in the knee muscles found that a heel hight of even three centimeters was enough to cause extra strain on the muscles, although unfortunately most studies do not focus on such small increments, preferring to focus their attacks on the known threat of high heels. Secondly, it is sometimes claimed that flip-flops will give you fallen arches, because they give you no arch support. This is actually backwards; wearing flip-flops appears to protect against the development of flat-footedness3. However, if you already have fallen arches, flip-flops will probably be uncomfortable.
References:
1. Effects of shoe heel height on biologic rollover characteristics (.pdf)
2. Effect of shoe heel height on vastus medialis and vastus lateralis electromyographic activity during sit to stand
3. The influence of footwear on the prevalence of flat foot (.pdf)