Narb is mostly right on this one - we do normally use fats as well as glucose for energy (in fact muscle tissue respires fats preferentially). However, your brain can ONLY respire glucose.
If you stop eating carbs on Atkins, or if you are diabetic so you have high blood glucose but are unable to take any of it up into cells (because you're not making insulin) then your brain needs an alternative source to glucose. Initially you make it by muscle breakdown and using certain amino acids to make glucose, but over a few weeks you start to turn on the enzymes that make ketones from fats (rather than just acetyl CoA which is what normal cells use but brain cells can't) - hence these are found in the blood and therefore urine. The brain gradually switches over to ketone metabolism as an alternative to glucose. However, because ketones are volatile and acidic, this is gradually poisonous and can't be sustained for long periods of time.