Covalent compounds are those which are made up by sharing of electrons between them and the electronegativity of the elements which form covalent compounds are nearly same that is they are generally non-polar in nature whereas ionic compounds are formed by complete transfer of electrons from one element to the other and thus resulting in formation of ions of opposite charges that is cation (with positive charge) and anion (with negative charge). There is a large difference in electronegativity of the element involved in the formation of ionic compounds. They are held together by electrostatic force of attraction between them.
The conductivity of a substance is determined by the flow of electric charge. When the charged particles move that are present in the ionic compounds only in the conductivity apparatus they complete the circuit by the flow of electrons.
Hence, a conductivity apparatus test whether a solution has ionic or covalent substances in it as ions complete the flow of electrons to form a circuit.