Per reader request, and in the interest of making this as easy as possible to reach the broadest amount of users, I've added a less technical version to the end of this post. If you start reading this, and cannot make heads or tails of it, please skip to the section at the bottom in italics, it should be far easier to understand, although it will miss some of the technical details that the first section has.
There are two separate and distinct classifications for fasting and/or underfeeding. Those two areas are short term fasting (< 3 days) and long term fasting (> 60 hours), here I'll describe both. See the link I posted above to learn more. There's a lot of research out there, I'm not going to cite it all, but it's available, and if you want to do more research please email me and I'll try to find what you are looking for.
The human body stores energy in the form of glucose, which is arranged in large groups called glycogen and is stored in an aqueous medium in the liver and to a lesser degree in muscle sites. Glycogen is what the mitochondria in cells use to facilitate energy production and Adenosine Triphosphate (ATP) production, which is the actual chemical the body uses for energy. While ATP can be generated from both glycogen directly or via the Krebs cycle (citric acid cycle) using pyruvate generated from protein and/or fats, glycogen is by far the preferred medium for this operation because of the amount of ATP it produces (far more than the Krebs cycle) and the speed at which this is accomplished. The other thing to keep in mind is that the brain, which uses about 20% of the energy from the Resting Metabolic Rate (RMR) cannot use fat as energy because of the size of lipoprotein molecules and the blood-brain barrier. Glucose or ketone bodies are required for brain and CNS energy. This will become important later so I thought I'd give it now as a baseline.
Short term fasting
STF means the lack of incoming energy for between 6 and 60 hours. In the initial stages of fasting, the body does not recognize fully the lack of fuel and continues using glycogen and fat (via lipolysis) for energy. During this time hormone production continues on as usual. In some cases the Active Metabolic Rate (AMR) can actually go up slightly causing a small increase in energy output. This is a good time to try for fat loss while maintaining a healthy energy balance in the body. If performing full fasts (no incoming energy), this state usually lasts for 24 to 36 hours, if performing extreme underfeeding in an adult with normal body fat levels (non-obese individuals) usually characterized by 600 to 1000 calorie per day or higher deficit and assuming micro-nutrient levels are sufficient; this state can last up to about 60 hours or longer. For obese individuals the rise in AMR is usually not seen, nor is the rise in lipolysis, but the state can last for weeks depending on the amount of available fat to be used for energy. The body will begin to prioritize the brain for glucose use, which is fine while glycogen levels are sufficient. Whole body energy can be maintained via the freeing of lipoprotein from adipose fat and the conversion of protein into pyruvate and ketone bodies.
Long Term Fasting
LTF begins when the body's glycogen levels are depleted. When this happens certain distinct changes in the body become evident. Urea nitrogen levels rise due to the increase in protein usage and fat sparring (nitrogen is a byproduct of protein conversion to usable energy in the Krebs cycle). Adipocyte cells begin producing higher levels of cortisol which is a hormone designed to mitigate the effectivness of insulin. With the rise in insulin resistance comes the reduction of blood sugar effectiveness and the subsequent slow down of RMR. In the short term, studies have noted an 8 to 15% decrease in RMR with longer term results reaching up to about 30% (in > 15 days). Other changes are a decrease in testosterone output and growth hormone levels, although it's not known as yet whether this is caused by the reduction in RMR or a reason for that reduction. Coupled with this change is the shift from a glucose burning metabolism to a fat and protein burning metabolism which focuses on the reduction of unused lean tissue (to help prevent unneeded energy waste that occurs from all lean tissue) and the increase in fat storage due.
Personal thoughts on this process and conclusions I glean.
While these facts are all from studies, the case is not yet closed on long term or short term fasting and more research is needed. But to date it's believed by myself and many others that while the term "Starvation Mode" is thrown around the nutrition community with reckless abandon, it's not a myth, it's simply misunderstood. People in the nutrition community need to make a better effort to understand how much energy is needed to stay out of LTF states, and allow the body to continue burning fat while minimizing the risk of excess protein burn.
There are at home tests available that can measure urea nitrogen levels, this can help in some cases to determine whether a person is using excess protein as energy and possibly help to keep a client or patient from entering the dreaded "starvation mode". Specifically for those people looking to lose small volumes of fat. Obese people have a larger window in which they can afford large calorie deficits without inflicting serious long term muscle and organ damage, but for those with only a modest amount of fat, that damage can begin in weeks, despite their best effort. And because of the insidious nature of energy deprivation, and the lack of easily readable signs, people will continue to use large deficits to lose small volumes, which is a losing proposition in the long term.
Less technical version.
Us humans use a sugar based solution called glycogen for energy to run our muscles and organs. This energy comes mainly from carbohydrates. And without going to deeply into it. We store this glycogen mix in our liver, and a little in our muscles.
In short term fasting, our body has enough stored energy to work normally for between 24 hours and about 60 hours, the exception being people with large fat stores (obesity), where the change can take as long as 15 or more days depending on how obese the person is, and a host of other factors. Essentially this means that you can fast(eat nothing) or undereat (eat less than maintenance calories) for a period of hours or days and not aversly affect your metabolic rate and not diminish how your body burns extra fat.
In long term fasting, the body has finally run low enough on stored energy that it begins reacting to the deficit. The body starts changing the hormones that control how, how much, and what we burn for energy in an attempt to conserve body fat (a famine response) and burn as little as possible. When this happens we burn less energy than normal (which is bad, it means we have less energy), our body begins to canabalize muscle tissue, and the immune system is less effective. Eventually, if it continues long enough, organ function can be compromised, and certain specific medical conditions can occur. We are not sure exactly when this process begins because of the extremely complex nature of the human metabolism and the difficulty in measuring in vivo (inside the body)chemical amounts during changing conditions (it's very inefficient to try to draw blood every 10 minutes for 2 days to see the changes in hormone levels).
My summation is basically saying that the concept commonly known as "starvation mode" actually represents long term fasting, and while the term is commonly misused, that doesn't make it a myth, it makes many people's perception wrong only. I also say that while these things I say are strongly correlated theories, they are not fact, as there is still more research needed to confirm these things. But in the abscense of more data, it's all we can go on.