When people eat less carbohydrate, their bodies turn to fat for energy, so it makes sense that more ketones are generated. Some of those ketones (acetoacetate and ß-hydroxybutyrate) are used for energy; the heart muscle and kidneys, for example, prefer ketones to glucose. Most cells, including the brain cells, are able to use ketones for at least part of their energy. But there is one type of ketone molecule, called acetone, that cannot be used and is excreted as waste, mostly in the urine and breath (sometimes causing a distinct breath odor).
If enough acetone is in our urine, it can be detected using a dipstick commonly called by the brand name Ketostix (though there are other brands, as well). Even though everyone is generating ketones continuously, this detection in the urine is what is commonly called "ketosis."
The higher the concentration of ketones in the urine, the more purple the sticks will turn. The Atkins Diet, in particular, advises people to monitor ketosis as an indication of fat burning. Other reduced carbohydrate diets don't pay much attention to this, or aren't low enough in carbs to make much of an impression on the sticks. (The latter type of diet is sometimes called a "nonketogenic" low-carb diet.)
Why do some people think ketosis is a bad thing?There is an assumption that if a body is burning a lot of fat for energy, it must not be getting "enough" glucose. However, there is no indication, from studying people on reduced carbohydrate diets, that this is the case (though there is usually a short period of adjustment -- less than a week, in most cases). Although it's true that our bodies can't break fat down into glucose (though, interestingly, they easily use glucose to make fat), our bodies can convert some of the protein we eat into glucose. Indeed, this works well for people who don't tolerate a lot of sugar, because this conversion happens slowly so it doesn't spike blood glucose.
A dangerous condition called ketoacidosis can develop in those with type 1 diabetes, and it is sometimes confused with normal ketosis. The body usually avoids this state by producing insulin, but people with type 1 diabetes are unable to produce insulin. Even most people with type 2 diabetes who inject insulin usually produce enough insulin of their own to prevent ketoacidosis.
Low Carb Luxury has a good primer on how to use Ketostix.