What is ketosis?
Ketosis is a natural metabolic state wherein your body runs on ketones rather than glucose for energy. It requires you maintain 0 or very few (typically under 50g per day absolute maximum) carbohydrates and only adequate levels of protein (your body can convert protein to glucose through gluconeogenesis, so overdoing protein will prevent a deep level of ketosis). High levels of fat are required to make up the calories and provide ketones.
Ketosis is also the same state your body enters when fasting or starving, which you can imagine there is an immense amount of research on. However the body is perfectly equipped for a regular ketogenic diet. The Inuit are an example of a culture who have been on a ketogenic diet for thousands of years. It's also been used to treat epilepsy with significant success. The induction phase of the Atkins diet is also ketogenic. There's a somewhat uncomfortable period of adaptation, but once adapted ketosis seems to have multiple benefits and the drawbacks seem easily mitigated with complete foods.
Ketosis is also sometimes confused with ketoacidosis, which is a dangerous diabetic condition. This is a misunderstanding; ketosis is not ketoacidosis.
A primary risk for a ketogenic diet is that avoiding carbohydrates makes it difficult to receive all the required nutrients, this is obviously not a problem for complete foods. The other main risk seems to be that it's easy to overdo protein if you are eating a lot of meat, which may be bad for your kidneys. This is also very easily controlled with complete foods. That said, please do your own research beyond this page before making a major change to your diet, and as with the rest of the wiki do not consider anything written here to constitute medical advice.
Drawbacks (mitigated after adaptation)
The other potential drawback for some people is lowered workout intensity due to there being no stores of glycogen (essentially glucose) in the body, however it would seem this is solved with a cyclic ketogenic diet (CKD): eat carbs one or two days every week or every other week, say on the weekend, or a targeted ketogenic diet (TKD): eat carbs before and after a workout. This keeps you in ketosis for the majority of the time. You need to adapt to ketosis before switching to one of these or you'll reset your adaptation progress. You can find more information about these variants elsewhere on the internet. It's also possible you won't notice reduced performance, there may be many factors.
Entering ketosis is quick on a zero-carbohydrate complete food formulation (a few days), however it takes time to become keto-adapted, or fat-adapted.
During this time you may experience symptoms of lethargy, headaches, light-headedness, weakness, nausea, and so forth. This seems understandable as your body is no longer receiving energy from carbohydrates and is not yet fully able to receive energy from fat. These symptoms may be greatly mitigated by extra/adequate electrolyte intake, Lyle McDonald suggests the following in addition to your normal intake: 3-5 grams extra sodium chloride (table salt), 1 gram potassium, 300 mg magnesium. Complete foods should be excellent for this. You can lose approximately 5 pounds of water weight almost immediately upon starting - it's important to realize that this initial rapid weight loss is in fact water, not fat.
The general recommendation seems to be to reduce carbohydrates to near zero immediately and hasten the transition rather than to step down gradually. Often it's only the first few days that are the most unpleasant, and far less so if you keep your electrolytes up. Between 1 and 8 weeks are usually cited for full adaptation, more commonly 2-3.
Coconut oil is high in MCTs (~60%). They are digested and burned as energy easily and in that respect bear some similarity to carbohydrates, but in a good way. They also generate more ketones than the long-chain triglycerides that compose the majority of dietary fat, and this means ketosis can be maintained with a slightly less restrictive carbohydrate and protein intake .
These are the commonly cited (anecdotal) benefits.
Significantly less hunger. The Atkins diet is often pitched as "eat unlimited protein and fat and still lose weight" (of course not technically true, but often true in practice).
Fat loss without calorie counting. Quite possibly due to less hunger, less caloric intake. However optimizing your body to burn fat may very well alter your energy expenditure. Here's an interesting study done with mice (I know, we're not mice), emphasis mice.
A high-fat, ketogenic diet induces a unique metabolic state in mice3
"Mice on [ketogenic diets] ate the same calories as mice on [control diets] and [high-fat high-carbohydrate diets], but weight dropped and stabilized at 85% initial weight, similar to [calorie restricted diets]. This was consistent with increased energy expenditure seen in animals fed [ketogenic diets] vs. those on [control diets] and [calorie restricted diets]."
Muscle-saving fat loss. In ketosis the body prefers ketones to glucose. With plenty of fat available to produce ketones there's no need to manufacture them through gluconeogenesis from protein.[citation available, somewhere]
Balanced energy. The glucose reserves in the body are quite small relatively speaking, somewhere around ~500 grams and require constant replenishing with carbohydrates. Fat storage on the other hand is quite large. When the body is adapted to burning fat as its primary fuel source it's easier for it to maintain a balance of blood-sugar and insulin. The result is you don't get sugar highs, and you don't get low on energy if you skip a meal or two.
Less sleep required, better sleep, waking more refreshed, etc. Ordinarily when we sleep, as when fasting, ketones are elevated in the blood. It could be postulated that when our body is highly adapted to burn fat and it's our primary source, more energy is available all night long, and waking is more natural.
Clearer skin (in some people). You can be skeptical of this one simply because there is very little research on acne and diet. The science behind this is that the insulin response to elevated blood sugar levels from carbohydrates is too high in some people (hyperinsulinemia) and this "elicits a hormonal response that supports unregulated epithelial growth and increased sebum secretion".
Other well-researched health benefits lower levels of the higher density (bad, phenotype B) LDL cholesterol, lower blood pressure, higher HDL (good) cholesterol. Also the ketogenic diet has a long history in effectively treating epilepsy when even other interventions have failed.
You can see a list of Ketogenic Recipes here.