In a experiment that dealt with electric conduction of many concentrated solutions, NaCl(s) created an unknown, to me, substance after adding a drop or two of water. Furthermore, it isn't the reaction of water and NaCl(s) that I don't understand, but the reaction between the anode and cathode when they came in contact with NaCl(aq), sparks were present and it seemed to have "burned" NaCl(s). The question is: what is the correct name for the part that "burnt"?
There is not "burning", one might see a spark. It is a well known phenomenon. Do you see lightening in the sky? Air is an insulator, yet when very high voltage is present it becomes a conductor. It is called dielectric barrier discharge. In the same way, discharge between two electrodes can occur in pure water (almost an insulator). Adding NaCl to water makes it a good conductor but still it is mostly water, and if the electrodes are close enough, electrical breakdown can occur. Instead of ions carrying the current as it should happen in an aqueous conducting solution, the current becomes electronic even in the presence of water. Apparently, the local heating between the electrodes is so high, that a gaseous plasma is formed between the electrodes and plasma are electrically conducting.
As suggested in the comment, always exercise cause caution and all experiments should be done under proper supervision by a trained person/teacher.