The Atkins diet is basically a modification of a ketogenice diet.  The brief excerpt from Wikipedia explains what a ketogenic diet is:

The ketogenic diet is a high fat, adequate protein, low carbohydrate diet, primarily used to treat difficult-to-control (refractory) epilepsy in children. The diet mimics aspects of starvation by forcing the body to burn fat rather than carbohydrate.

The ketogenic diet was derived from research done on fasting diets.  Clinical trial were performed as early as 1911 and evolved into the ketogenic diet in the 1920's. 

The Atkins diet and other low-carbohydrate diets are not exactly a ketogenic diet, but a modified ketogenic diet.  The idea of this particular type of diet is to restrict carbohydrates so that the body go into a state of ketosis in which the liver converts fat into fatty acids which the body can burn as fuel more readily.

It is essential to restrict the carbohydrate intake in order for the body to enter the state of ketosis as carbohydrates are converted to glucose which is a more easily burned energy source.  Having the carbohydrates in you diet puts the fat at the back of the line so to speak - carbs burned first, and then if you still are burning calories, then the fats will start to burn off.

The low carb diets are restrictive and you are likely to ingest fewer calories resulting in weight loss.   So why not just eat fewer calories on a normal diet?  The reason being is the a low carb diet leaves you feeling more satiated - probably due to the longer time it takes protein and fats to be digested by the body.

So what do you eat?

Meat, cheese, eggs - things that don't have carbohydrates in them - simple!  Obviously, this will sound good at first to many people, but it gets monotonous rather fast.  I know some of you are thinking that you'll have BBQ and hamburger with gravy, and all kind of good stuff - problem is that most sauces are out - the food sources are relatively basic. 

The Atkins diet which I am going to reference for simplicity (but it is not the only low carb diet regimin) is actually broken down into four phases - Induction, Ongoing Weight Loss, Pre-Maintenance, and Lifetime Maintenance.

The Induction portion is where your body goes into ketosis by limiting carbohydrate intake to less than 20 grams (actually 15-20 grams).  You get these in the form of vegetable...but 15-20 grams really is not a lot - you must measure out your portions.

After the initial phase, you get to introduce some new food items and increase carbohydrate intake - but still not a lot.  If you are interested in following this diet, there are many resources out there that can give you the full run-down - I suggest you read these three entries in Wikipedia before you decide whether or not you want to pursue this diet - they will give you most of the information that you need to make an educated decision: Atkins Diet, Ketogenic Diet, Ketosis