What is a functional group?
A group of atoms in a molecule that introduces functionality into the species.
Functionality means many things, but most importantly, reactivity and interactivity.
By reactivity I mean asking: What reactions will this molecule undergo?
By interactivity I mean asking: What non-covalent interactions will this molecule have with others?
The former is more important in organic chemistry and the latter is more important in biological chemistry. Nevertheless, both are significant to both disciplines.