When a solute dissolves into the solution without disruption of its molecular (covalent) structure, it is said to be a non-electrolyte and the resulting solutions does not conduct electricity.
When many soluble ionic compounds are dissolved in a polar solvent, the compound completely dissociates to form ions, which results in a solution than can conduct electricity well. This is called a strong electrolyte.
If a solute dissociates somewhat to form ions, but not completely, it is called a weak electrolyte. Weak acids and bases are weak electrolytes.

Hydration is the process by which the solvent (water) disrupts the lattice of an ionic solid and forms dissolved ions that are surrounded by a shell of associated water molecules (hydration shell). If the solvent is not water we call this 'solvation' and the resulting structure a 'solvation shell'.

The difference between an electrolyte and non-electrolyte solution can be seen in this animation.