Ketogenic diets have been around for a long time -- and I think there is a recent resurgence in interest for a number of reasons. A ketogenic diet is defined as consuming somewhere between 100 and 0 grams of carbs per day (most information sources agree that less than 50 grams of carbs per day is achievable and will put most people in ketosis in anywhere from 3 to 20 days. It takes me about 3 days.
The actual suggested percentages can vary depending on the information source you draw from, but it goes something like this: 70% of calories from fat, 25% protein, and 5% carbs. There is evidence monounsaturated polyunsaturated fat is more ketogenic than saturated fat -- but saturated fat will still get you there. (Differential Metabolic Effects of Saturated Versus Polyunsaturated Fats in Ketogenic Diets, Fuerhlein, et al., 2004).
Everyone is different, so your mileage may vary. There is an unofficial writeup on another forum (with food suggestions). (Try googling: A guide to ketosis).
They have been prescribed to people with epileptic seizures, and it has been clinically shown to reduce seizures in a sub-set of patients. There is "strong supportive evidence" in clinical trials that it is an effective weight-loss diet. (Beyond weight loss: a review of the therapeutic uses of very-low-carbohydrate (ketogenic) diets, Paoli et al., 2013).
The basic idea is that by eating so few carbs, your body switches over to a mechanism where ketones are produced through a lipolysis process (I'm not a biochemist, so you have to do your own reading if you want those details). As long as you are not diabetic, it is thought to be relatively safe to try (do not confuse ketosis with ketoacidosis -- a dangerous condition for diabetics).
Many sources of information will talk about a "mental clarity" that is achieved on ketosis, but it is all anecdotal, and I'm not aware of any studies that have prospectively examined this claim (so I'm skeptical). (The effects of the ketogenic diet on behavior and cognition, Hallbook, et al., 2012). I've been in ketosis, and I agree that "you feel different", but I wouldn't describe it as mental clarity.
Currently, there are some clinical trials investigating whether a ketogenic diet can slow cancer growth, and/or improve the effects of chemo on cancers. Some types of cancer are very glycolytic, having as much as 50x the glucose uptake of normal cells, so it stands to reason that a low glucose environment in the body may be helpful. However, not all cancers cells are this way, so if it does work, it will only work on a few types of cancers.
It will reduce your blood glucose and blood insulin levels significantly (Beyond weight loss: a review of the therapeutic uses of very-low-carbohydrate (ketogenic) diets, Paoli et al., 2013).
It has been proposed to benefit/prevent alzheimer's and parkinson's patients (as a neuroprotectant) and cardiovascular disease -- but I'm not aware of any data to support those suppositions.
You can buy "ketosticks" to test for ketone bodies in urine (50 strips costs about $10 or less). There are also blood meters (very similar to glucose meters) that can be had for less than $50, with strips costing about $1 per strip (pretty pricey, IMO).
Your first bowl of oats probably puts you over the 50 g's of carb limit.
Typical ketogenic diet is: oils, meats, butter, creams, cheese, nuts, veggies, limited fruit. Stay away from breads, flour, pasta, sugar...
Also note, too much protein will supposedly kick you out of ketosis too, but everyone is different, and you'd need to experiment.