In everyday English, when someone mentions sugar, they are probably talking about sucrose aka 'table sugar'. And you are correct that sucrose consists of a glucose and fructose molecule bonded together.

In a more...official or scientific context, "Sugar is the generalized name for sweet, short-chain, soluble carbohydrates, many of which are used in food." as Wikipedia puts it. That means that sucrose, glucose and fructose (and many others) are all considered to be sugar.

Brain cells can use ketones instead of glucose. That's what happens on a diet that severely restricts carbohydrates.