Coenzyme, Cofactor
 
The term cofactor has been used as a general term to indicate that a compound was required, in addition to the enzyme, for a reaction to proceed and that the compound remained unchanged at the end of the reaction. Cofactors were considered to be either activators, such as metal ions, or coenzymes , organic molecules that participated in enzymic reactions. NAD and NADP were named originally as Coenzyme I (CoI) and Coenzyme II (CoII), respectively, because in studies on metabolic reactions they appeared to function as electron carriers. They certainly perform that role in coupled reactions, such as those catalyzed by glyceraldehyde-3-phosphate dehydrogenase and lactate dehydrogenase: 

However, this is not the role of NAD/NADH when functioning as substrates for a dehydrogenase reaction. Similar comments can be made about Coenzyme A (CoA), which functions as an acyl carrier, and Coenzyme Q (CoQ, ubiquinone), which acts as an electron carrier. In this connection, it is of interest that nucleoside phosphates, such as ATP, are not usually classified as carriers of  phosphoryl groups.
Other compounds that have been considered as coenzymes are biotin, flavin adenine di- and mono-nucleotide (FAD, FMN), lipoic acid, pyridoxal phosphate, and thiamin pyrophosphate. Biotin, lipoic acid, and pyridoxal phosphate are bound covalently to carboxylases, acyl transferases, and aminotransferases, respectively. FAD may, or may not, be bound covalently to oxidases. The interaction of thiamin pyrophosphate with decarboxylases is noncovalent, although the binding is tight.
It would seem preferable to consider a compound as a coenzyme only under conditions where it is functioning as a carrier and not when it is simply a substrate for a single enzyme. Compounds that are covalently or tightly bound to an enzyme might better be regarded as prosthetic groups. The features that these compounds have in common are the helping groups that they provide to enzymes. These groups possess chemical attributes that the enzymes do not have and that are essential for catalysis.