Ionic compounds have giant structures in which many strong electrostatic forces hold the ions tightly together. This means they are solids at room temperature. Ionic compounds have high melting points and high boiling points.
When they have melted they are free to move. This allows them to carry electrical charge, so the liquids conduct electricity. Some ionic solids dissolve in water because the water molecules can split up the lattice. The ions are free to move in the solutions and so they also conduct electricity.