If calories are a measure of energy, and protein contains more calories per gram than carbohydrates then how does Atkins work? What is the difference between the energy provided by the 'carbs' and the 'calorie' energy that is in all food? Why does eating no carbohydrates make your body go into 'fat burning' mode for energy if everything contains calories?
Your premise is a bit off. Protein and Carbohydrates have essentially the same number of Calories. It's fat that has more calories.
In order to understand how Atkins and other ketogenic diets work, you need to understand a bit about energy systems and a very important pair of hormones. Oversimplifying things quite a bit, we can get to a layman's understanding of what's going on. Essentially, there are three sources of energy the body can pull from:
Under normal operations, your body will use up the glycogen before it attempts to use any other source of energy. The problem is when we have depleted the glycogen, the body tends to consume both the muscle/organs and fat stores at the same time--unless we do something different.
For the average person, your body will hold about 3 days of glycogen energy. Glycogen is replenished by carbohydrates. In a ketogenic diet, you are not eating carbohydrates and are intentionally burning through your glycogen stores to force your body into a state where it creates ketone bodies using the available energy. The ketone bodies cannot be re-assimilated as fat, and have to be urinated out. This is a good thing. However, remember how we said the body will pull from both the muscles/organs and fat stores if there is no glycogen? Here's what the body gets from those sources of energy:
If you have a Caloric deficit of 1200 Calories a week, and the body is pulling half from fat and half from muscle tissue, you will burn a pound of muscle while only burning less than a quarter of a pound of fat. This isn't good. Again, this is also grossly oversimplified. Bottom line is that we have to do something to protect the muscle.
This is where the high protein content of your diet comes to play. The body doesn't really care if the protein it is consuming is dietary or from your existing muscles/organs. It's just going to create ketone bodies using both the protein and the fat. Sufficiently high protein, and you can actually gain some muscle mass. An additional benefit of protein is that it is very thermogenic, or in other words it takes a lot of energy to process. You burn more Calories digesting protein than you do carbohydrates and fat.
NOTE: part of the process of digesting protein requires calcium. If you don't have enough dietary or supplementary calcium, you may end up leaching that calcium from your bones. This is particularly troublesome for women as they have a greater risk of osteoporosis anyway.
A big influence on the way your body processes energy has to do with a pair of hormones: insulin and glucagon. Insulin is anabolic, which means it's job is to build up. It operates by taking elevated blood sugar and pushing it to the muscles and organs. However, if both the muscles and organs are full, the only place left is fat cells. Glucagon has the opposite responsibility, and is catabolic, meaning it's job is to tear down. It operates to restore blood sugar to normal levels by taking energy out of the muscles, organs, and fat cells. The energy in muscles and organs is glycogen. The energy in fat cells is fat.
In a ketogenic diet, you are depleting the glycogen stores so that the only place left to pull energy is the fat stores. There are a few areas of concern here, which affect active people more than sedentary people:
I'm not an advocate for Atkins but the theory is that carbohydrates, proteins and fat are all metabolised by the body in different ways before it becomes the direct food that our cells use. Low carbs advocates suggest that digesting carbohydrates increases the level of the hormone insulin in our blood leading to cells accepting that energy and storing it as fat.
Conversely, eating low levels of carbs does not raise insulin and thus that storage mechanism is not invoked. The body also has to switch processes to metabolise the fat and protein you're eating - which is pretty much the same mechanism it uses to burn body fat.
Secondly, there's some evidence that it's the protein in food that makes us feel full and stop eating. There's certainly evidence that Atkins is simply a low-calorie diet. Theories are that this is because it's rather hard to find food that is low carb and because of the satiation I just mentioned.