Ionic solids are brittle but very strong. They do not conduct electricity but do when molten or in solution. They have high melting points. Molecular solids, on the other hand, have much lower melting points and will not conduct electricity even when molten or in solution.
All of these properties can be explained when we look at the particles and the forces (bonds) which hold these together. These are covered in a few videos my classes made of me teaching, but I will also describe the key points below.
Ionic substances are (not surprisingly) made up of ions. Ions are charged atoms (or groups of atoms). They are charged because they have gained or lost electrons to become more stable. Cations and anions are the names given to positive and negative ions, respectively.
Cations and anions are held together by an electrostatic attraction called an ionic bond. Ionic bonds are very strong which explains the strength and melting/boiling points of ionic substances.
As ions are charged, we would expect ionic substances to conduct electricity. However, the ionic bond is too strong to allow for the movement of ions while in the solid state. If these bonds are broken, the ions are free to move so a current may flow. This is why molten ionic substances (and solutions) do conduct electricity.
Molecular substances, on the other hand, are made up of electrically-neutral particles called molecules. These are held together by very weak inter-molecular forces, called dispersion forces or van der Waal's forces.
The bonding between the atoms in a molecule is covalent bonding, which is strong. This is why the particles themselves (molecules) are harder to break apart into their atoms again. However, this plays no part in understanding the physical properties of molecular substances.