Most nonelectrolytes are covalently bonded organic compounds with very stable internal structures. Sugars, starches, and rubbing alcohol are very common everyday examples of these chemicals. When you add one of these substances to a solvent like pure water, it dissolves.
The water molecules easily pull the individual molecules of the substance away from each other. However, the strong covalent bonds holding each individual molecule together remain completely intact. The dissolved molecules absolutely do not split apart into positive and negative pieces.
Electricity requires freely moving charged particles in order to flow through a liquid solution. Because a nonelectrolyte produces only neutral molecules, the liquid cannot carry an electric charge. Students sometimes think that anything that dissolves easily must also conduct electricity.
This is a very common and completely understandable misconception in chemistry classes. Solubility simply means the chemical molecules can mix evenly into the surrounding water. Conductivity specifically requires the substance to break apart into freely moving charged ions. If a substance does not form ions, it is a nonelectrolyte regardless of its solubility.