Although one might naïvely imagine that the term carbohydrate is used for a molecule containing carbon, hydrogen and oxygen, this is not the case as, if so, it would include compounds like fatty acids. According to the Wikipedia entry it is more restricted, although still somewhat imprecise:
A carbohydrate is a biological molecule consisting of carbon (C), hydrogen (H) and oxygen (O) atoms, usually with a hydrogen–oxygen atom ratio of 2:1 (as in water); in other words, with the empirical formula Cm(H2O)n (where m could be different from n). Some exceptions exist; for example, deoxyribose, a sugar component of DNA has the empirical formula C5H10O4. Carbohydrates are technically hydrates of carbon; structurally it is more accurate to view them as polyhydroxy aldehydes and ketones.
By this definition all sugars are carbohydrates if one accepts the authoratitive definition of a sugar by Ron on SE Chemistry:
Simple sugars have the molecular formula Cn(H2O)n, where n is at least 3. They also have to be capable of forming an aldehyde or ketone carbonyl group.
He uses the qualifier of being able to form an aldehyde or ketone to exclude certain carbohydrates such as inositol from being classified as sugars:
…compounds in the inositol family [that] fit the molecular formula requirement, but are not considered to be sugars because they are incapable of forming a carbonyl.
So, yes, all sugars are carbohydrates, but not all carbohydrates are sugars. (Which, despite the unsatisfactory nature of the definitions, is what is generally accepted.)
This answer describes the terminology used in biological sciences. Because it exists one has to work with it, but one should realize that it is not chemically exact, as illustrated by the following.
Chemists (and biochemists) refer to reducing sugars — and hence by implication there must be non-reducing sugars. The disaccharide sucrose — which is what most of us think of as sugar — is a non-reducing sugar. But the reducing property is a feature of the aldehyde or ketone group, which by Ron’s definition (which I emphasize is that accepted by biologists) would exclude sucrose from being a sugar! Clearly there is a conflict here.
The Wikipedia entry for Reducing Sugars includes the following:
A reducing sugar is any sugar that is capable of acting as a reducing agent because it has a free aldehyde group or a free ketone group. All monosaccharides are reducing sugars, along with some disaccharides, oligosaccharides, and polysaccharides.
Disaccharides are formed from two monosaccharides and can be classified as either reducing or nonreducing. Nonreducing disaccharides like sucrose and trehalose have glycosidic bonds between their anomeric carbons and thus cannot convert to an open-chain form with an aldehyde group; they are stuck in the cyclic form. Reducing disaccharides like lactose and maltose have only one of their two anomeric carbons involved in the glycosidic bond, meaning that they can convert to an open-chain form with an aldehyde group.
The way out of the dilemma available to chemists (but not to dieticians) would seem to be not to talk about sugars and carbohydrates at all, but refer to specific chemical types: monosaccharides, disaccharides etc. and their derivatives; and for other dietary/storage molecules refer to fatty acids, triglycerides etc.