Do most organic compounds contain carbon?

Yes, in order to be classified organic, the compounds have to contain carbon. This however must not be confused with the use of the term “organic” in popular terminology in which "organic" means natural and healthy. If we are discussing scientific terminology, as per that, an organic compound that comprises of one or more atoms of various elements will contain carbon in various combinations with elements like oxygen, hydrogen, sulphur and nitrogen etc.
All organic compounds need to contain carbon to be considered organic. Identifying such compounds is a science in itself. Organic compounds are identified firstly by the fact that they have carbon. They are also identified by chemical reaction for instance how they are broken down (in cases like proteins being broken down or amides in the digestive system). They are also identified by the molecular structure. These are the various ways to identify organic compounds. It can be a bit confusing for the lay man who considers water organic since it comes from natural sources, but since it does not contain carbon, it is not classified as organic in chemistry.

Another important component in an organic compound is hydrogen. Actually a compound needs to contain both hydrogen and carbon for it to be considered organic. Therefore carbon dioxide is not organic, because even though it contains carbon it does not contain hydrogen.

Some instances of organic compounds are glucose, fructose, nicotine, caffeine, cocaine.

A nutrient may or may not be an organic compound. An excellent example of this is the fact that plants do not need any organic compound for their nutrition. They make their own nutrients using inorganic material like water, carbon dioxide, minerals and even oxygen. However human beings need both organic and inorganic nutrients. To be a nutrient, the compound needs to be biodegradable. It needs to be broken down by the living organism into smaller parts that can then be absorbed by the body for its own food.

Let us consider the case of proteins, lipids, carbohydrates, and nucleic acids. These are valuable nutrients that are used by the body. Now all the above mentioned compounds are biological compounds and are biodegradable and are absorbed by the human body. All organic compounds are biodegradable and therefore do not disturb the ecological balance of the planet. Even oils, alcohols, etc are biodegradable. Often all that is needed is the right sort of enzyme to break them into smaller particles for easy absorption.