[jcranmer]

There's no terribly good definition of organic molecules in chemistry. The traditional definition is something that required a living organism to produce it, until inorganic synthesis of organic urea ruined that idea. The typical rule of thumb is an organic compound contains C-C and C-H bonds, but there's a whole host of exceptions to that rule (including urea, which lacks both C-C and C-H bonds).

The best I can come up with is that organic chemistry is the study of the interactions of a set of common structural motifs (called functional groups), and an organic molecule is something that contains those functional groups.