Does Distilled Water Conduct Electricity?

*Everything* conducts electricity to some degree; there are no perfect electrical insulators just as there are no perfect thermal insulators.  Materials that are uses as insulators are often called "dielectrics".
Water as we generally experience it conducts electricity fairly well because of the free ions dissolved in it.  Salt has lots of ions to add, and salt water is a good conductor.  Tap water conducts well as there are chlorine ions and all kinds of other stuff you don't want to know about.

Distilled water has few ions and is not very conductive, but usually has enough ions to register measurable conductivity.

Now, deionized water, is water that has had all (or at least measurably so) of the ions remove through various mechanical means, starting with reverse osmosis.  DI water is such a poor conductor of electricity that it can be, and has been, used as a dielectric.